<Text-field style="Heading 1" layout="Heading 1">Inicializa\303\247\303\265es</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Solucionando o Problema de Autovalor de Sturm-Liouville</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Condi\303\247\303\265es Homog\303\252neas de Dirichlet</Text-field>

Um probelma de autovalor de Sturm-Liouville surge se unirmos a equa\303\247\303\243o diferencial

JkkjREVHNiI2IyIiIg==

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiWEc2IjYjSSJ4R0YsRi5GLiIiIiomSSdsYW1iZGFHRixGL0YqRi8hIiIiIiE=

com as condi\303\247\303\265es de fronteira homog\303\252neas de Dirichlet 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 Esse \303\251 um exemplo de condi\303\247\303\265es "separadas" para as quais garantimos um conjunto completo de autofun\303\247\303\265es ortogonais. Uma abordagem para solucionar esse problema de autovalor pode ser invocar o comando do Maple dsolve, com o resultado

LUknZHNvbHZlRzYiNiQ8JSZJI0RFR0YkNiMiIiIvLUkiWEdGJDYjIiIhRi8vLUYtNiNJI1BpRyUqcHJvdGVjdGVkR0YvLUYtNiNJInhHRiQ=

Ly1JIlhHNiI2I0kieEdGJSIiIQ==

Infelizmente, o Maple retorna a solu\303\247\303\243o-zero, e n\303\243o sequer come\303\247a a resolver o problema de autovalor. \303\211 por isso que estamos escrevendo essa s\303\251rie de artigos. O Maple pode atualmente ser de grande ajuda na solu\303\247\303\243o de problemas, mas precisa ser guiado pelo usu\303\241rio.

A equa\303\247\303\243o caracter\303\255stica para \303\251 obtida fazendo a suposi\303\247\303\243o exponencial

PkkjRUdHNiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSJtR0YkIiIiSSJ4R0YkRi0=

LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkibUdGJyIiIkkieEdGJ0Yr

Substitui\303\247\303\243o e \303\241lgebra \303\263bvia nos d\303\243o

KiZJI0VHRzYiISIiLUkoY29sbGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUklZXZhbEdGKTYkJkkjREVHRiQ2IyIiIi8tSSJYR0YkNiNJInhHRiRGI0kkZXhwR0YoRjI=

LywmKiRJIm1HNiIiIiMiIiJJJ2xhbWJkYUdGJiEiIiIiIQ==

Deveria ser \303\263bvio que as ra\303\255zes caracter\303\255sticas s\303\243o 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, e que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= \303\251 o caso de "raiz repetida". Desde a forma da solu\303\247\303\243o geral de muda para esse caso, consideramos primeiro um caso especial. A equa\303\247\303\243o diferencial \303\251 ent\303\243o

PklGI21zdWIobWkoIkRFIiksbXJvdyhtbigiMSIpLG1pKCJhIikpKUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQmSSNERUdGJDYjIiIiL0knbGFtYmRhR0YkIiIh

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JIlhHNiI2I0kieEdGK0YtRi0iIiE=

e uma solu\303\247\303\243o para essa equa\303\247\303\243o com somente a condi\303\247\303\243o de limite \303\240 esquerda \303\251

LUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JElGI21zdWIobWkoIkRFIiksbXJvdyhtbigiMSIpLG1pKCJhIikpKUdGJy8tSSJYR0YnNiMiIiFGLy1GLTYjSSJ4R0Yn

Ly1JIlhHNiI2I0kieEdGJSomSSRfQzFHRiUiIiJGJ0Yq

Aplicando a condi\303\247\303\243o delimite \303\240 direita a essa solu\303\247\303\243o leva \303\240 equa\303\247\303\243o

Ly1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSRyaHNHRiU2Iy8tSSJYRzYiNiNJInhHRi0qJkkkX0MxR0YtIiIiRi9GMi9GL0kjUGlHRiUiIiE=

LyomSSRfQzFHNiIiIiJJI1BpRyUqcHJvdGVjdGVkR0YmIiIh

da qual \303\251 claro que a constante multiplicativa tem que ser zero. Desde que isso leve \303\240 solu\303\247\303\243o trivial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRj1GPQ==, conclu\303\255mos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= n\303\243o \303\251 um autovalor.

Consideremos agora o caso LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyLUY2Ni1RIixGJ0YyRjkvRjxRJXRydWVGJ0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktRLDAuMzMzMzMzM2VtRidGMg==e novamente usamos a estrat\303\251gia de aplicar somente o limite \303\240 esquerda. A solu\303\247\303\243o resultante \303\251

Pkk3I21zdWIobWkoIlgiKSxtbigiMSIpKUc2Ii1JJHJoc0clKnByb3RlY3RlZEc2Iy1JJ2Rzb2x2ZUc2JEYnSShfc3lzbGliR0YkNiQ8JCZJI0RFR0YkNiMiIiIvLUkiWEdGJDYjIiIhRjctRjU2I0kieEdGJA==

LCYqJkkkX0MyRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkknbGFtYmRhR0YlI0YmIiIjSSJ4R0YlRiZGJiEiIiomRiRGJi1GKDYjLCRGLUYyRiZGJg==

Como o Maple pode retornar essa solu\303\247\303\243o com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw== como a constante arbitr\303\241ria, usamos o seguinte dispositivo para reescrever a solu\303\247\303\243o em termos da constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. N\303\243o importa que nome o Maple retorne para a constante arbitr\303\241ria, nossa constru\303\247\303\243o ser\303\241 sempre em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

QyQ+SSZwYXJhbUc2IiYtSSdyZW1vdmVHJSpwcm90ZWN0ZWRHNiVJJGhhc0dGKS1JJ2luZGV0c0dGKTYjSTcjbXN1YihtaSgiWCIpLG1uKCIxIikpR0YlPCRJJ2xhbWJkYUdGJUkieEdGJTYjIiIiISIi Pkk3I21zdWIobWkoIlgiKSxtbigiMiIpKUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJNyNtc3ViKG1pKCJYIiksbW4oIjEiKSlHRiQvSSZwYXJhbUdGJEkiY0dGJA==

LCYqJkkiY0c2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJJ2xhbWJkYUdGJSNGJiIiI0kieEdGJUYmRiYhIiIqJkYkRiYtRig2IywkRi1GMkYmRiY=

Aplicando a condi\303\247\303\243o de limite \303\240 direita leva a equa\303\247\303\243o alg\303\251brica

Ly1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJNyNtc3ViKG1pKCJYIiksbW4oIjIiKSlHNiIvSSJ4R0YoSSNQaUdGJSIiIQ==

LywmKiZJImNHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2IyomSSdsYW1iZGFHRiYjRiciIiNJI1BpR0YrRidGJyEiIiomRiVGJy1GKTYjLCRGLkYzRidGJyIiIQ==

sem interven\303\247\303\243o, o Maple providenciar\303\241 somente a solu\303\247\303\243o zero para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMg==como confirmamos com

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8sJiomSSJjR0YnIiIiLUkkZXhwR0YkNiMqJkknbGFtYmRhR0YnI0YtIiIjSSNQaUdGJUYtRi0hIiIqJkYsRi0tRi82IywkRjFGNkYtRi0iIiFGMg==

IiIh

A ferramenta necess\303\241ria \303\251 a vari\303\241vel de ambiente

PkkxX0VudkFsbFNvbHV0aW9uc0c2IkkldHJ1ZUclKnByb3RlY3RlZEc=

SSV0cnVlRyUqcHJvdGVjdGVkRw==

o que, se estabelecida o valor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEodmVyZGFkZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, permite ao Maple retornar a solu\303\247\303\243o

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8sJiomSSJjR0YnIiIiLUkkZXhwR0YkNiMqJkknbGFtYmRhR0YnI0YtIiIjSSNQaUdGJUYtRi0hIiIqJkYsRi0tRi82IywkRjFGNkYtRi0iIiFGMg==

LCQqJEklX1oxfGlyRzYiIiIjISIi

Maple usa a nota\303\247\303\243o 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 indicar um inteiro. Infelizmente, cada vez que esse comando solve \303\251 executado, o \303\255ndice LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= ir\303\241 incrementar por 1, e novamente, n\303\243o podemos referenciar esse s\303\255mbolo em uma maneira segura. Desse modo, recorremos ao seguinte dispositivo

QyQ+SSV0ZW1wRzYiJi1JJ2luZGV0c0clKnByb3RlY3RlZEc2IywkKiRJJV9aMXxpckdGJSIiIyEiIjYjIiIiRi8= LUklZXZhbEclKnByb3RlY3RlZEc2JCwkKiRJJV9aMXxpckc2IiIiIyEiIi9JJXRlbXBHRilJIm5HRik=

LCQqJEkibkc2IiIiIyEiIg==

para escrever os autovalores com uma nota\303\247\303\243o \303\272nica. Os autovalores s\303\243o os n\303\272meros negativos 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

Tamb\303\251m \303\251 poss\303\255vel conduzir o Maple atrav\303\251s de passos resultando nessa solu\303\247\303\243o para os autovalores. Come\303\247amos escrevendo 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 na forma

LUknZmFjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvLCYqJkkiY0dGJyIiIi1JJGV4cEdGJDYjKiZJJ2xhbWJkYUdGJyNGLSIiI0kjUGlHRiVGLUYtISIiKiZGLEYtLUYvNiMsJEYxRjZGLUYtIiIh

LywkKiZJImNHNiIiIiIsJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjKiZJJ2xhbWJkYUdGJiNGJyIiI0kjUGlHRixGJ0YnLUYqNiMsJEYvISIiRjdGJ0Y3IiIh

Uma vez que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= poderia resultar na solu\303\247\303\243o trivial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEiMEYnRj1GPQ==, devemos ter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=, e ent\303\243o

LUkkbWFwRyUqcHJvdGVjdGVkRzYlSSZjb2VmZkdGJC8sJComSSJjRzYiIiIiLCYtSSRleHBHNiRGJEkoX3N5c2xpYkdGKzYjKiZJJ2xhbWJkYUdGKyNGLCIiI0kjUGlHRiRGLEYsLUYvNiMsJEYzISIiRjtGLEY7IiIhRio=

LywmLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkknbGFtYmRhR0YpIyIiIiIiI0kjUGlHRidGLiEiIi1GJTYjLCRGK0YxRi4iIiE=

Essa diferen\303\247a de exponenciais sugerem reescrever o lado esquerdo como

LUkoY29udmVydEclKnByb3RlY3RlZEc2JC8sJi1JJGV4cEc2JEYkSShfc3lzbGliRzYiNiMqJkknbGFtYmRhR0YsIyIiIiIiI0kjUGlHRiRGMSEiIi1GKTYjLCRGLkY0RjEiIiFJJXRyaWdHRiw=

LywkLUklc2luaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJ2xhbWJkYUdGKSMiIiIiIiNJI1BpR0YnRi4hIiMiIiE=

para o qual realizamos que se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNiQtRiw2JlEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y9Rj0= \303\251 real, a \303\272nica solu\303\247\303\243o para essa equa\303\247\303\243o ser\303\241 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyLUY2Ni1RIixGJ0YyRjkvRjxRJXRydWVGJ0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktRLDAuMzMzMzMzM2VtRidGMg==novamente conduzindo \303\240 solu\303\247\303\243o trivial. Isso inspira a substitui\303\247\303\243o

LUkpYXNzdW1pbmdHNiI2JDcjLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JJmV2YWxjR0YkNiMtSSVldmFsR0YqNiQvLCQtSSVzaW5oR0YpNiMqJkknbGFtYmRhR0YkIyIiIiIiI0kjUGlHRipGOyEiIyIiIS9GOSwkKiRJI211R0YkRjwhIiI3Iy1JIj5HRio2JEZDRj8=

LyomXiMhIiMiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSNtdUdGLEYmSSNQaUdGKkYmRiYiIiE=

do qual segue facilmente que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 um inteiro, dado pelo Maple por

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8qJl4jISIjIiIiLUkkc2luR0YkNiMqJkkjbXVHRidGLUkjUGlHRiVGLUYtIiIhRjI=

SSVfWjJ8aXJHNiI=

<Text-field style="Heading 2" layout="Heading 2">Condi\303\247\303\265es Homog\303\252neas de Neumann</Text-field>

O problema de autovalor de Sturm-Liouville aparece quando as condi\303\247\303\265es homog\303\252neas (e separadas) de fronteira de Neumann 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 s\303\243o aplicadas \303\240 equa\303\247\303\243o diferencial

JkkjREVHNiI2IyIiIg==

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiWEc2IjYjSSJ4R0YsRi5GLiIiIiomSSdsYW1iZGFHRixGL0YqRi8hIiIiIiE=

ter\303\241 um conjunto diferente de autovalores e autofun\303\247\303\265es do que obtemos com as condi\303\247\303\265es de Dirichlet. Desde que j\303\241 saibamos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= deve ser manuseado como um caso especial, come\303\247amos obtendo a solu\303\247\303\243o geral da EDO

SUYjbXN1YihtaSgiREUiKSxtcm93KG1uKCIxIiksbWkoImEiKSkpRzYi

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JIlhHNiI2I0kieEdGK0YtRi0iIiE=

Novamente tomamos precau\303\247\303\265es para escrever a solu\303\247\303\243o com constantes arbitr\303\241rias \303\272nicas, obtendo

QyQ+SSV0ZW1wRzYiLUkkcmhzRyUqcHJvdGVjdGVkRzYjLUknZHNvbHZlRzYkRihJKF9zeXNsaWJHRiU2JElGI21zdWIobWkoIkRFIiksbXJvdyhtbigiMSIpLG1pKCJhIikpKUdGJS1JIlhHRiU2I0kieEdGJSEiIg== Pkk3I21zdWIobWkoIlgiKSxtbigiMCIpKUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJJXRlbXBHRiQ3JC9JJF9DMUdGJEkiYkdGJC9JJF9DMkdGJEkiYUdGJA==

LCYqJkkiYkc2IiIiIkkieEdGJUYmRiZJImFHRiVGJg==

Para essa solu\303\247\303\243o geral, aplicamos as duas condi\303\247\303\265es limite, resultando nas equa\303\247\303\265es alg\303\251bricas

Ly1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmR0YlNiRJNyNtc3ViKG1pKCJYIiksbW4oIjAiKSlHNiJJInhHRisvRiwiIiFGLg==

L0kiYkc2IiIiIQ==

Ly1JJWV2YWxHJSpwcm90ZWN0ZWRHNiQtSSVkaWZmR0YlNiRJNyNtc3ViKG1pKCJYIiksbW4oIjAiKSlHNiJJInhHRisvRixJI1BpR0YlIiIh

L0kiYkc2IiIiIQ==

Consequentemente, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= \303\251 um autovalor, e a autofun\303\247\303\243o correspondente \303\251 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

Agora, condere o caso LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIzAuRidGMkYy Novamente encontramos a solu\303\247\303\243o geral, e a escrevemos com constantes arbitr\303\241rias nomeadas unicamente.

QyQ+SSV0ZW1wRzYiLUkkcmhzRyUqcHJvdGVjdGVkRzYjLUknZHNvbHZlRzYkRihJKF9zeXNsaWJHRiU2JCZJI0RFR0YlNiMiIiItSSJYR0YlNiNJInhHRiUhIiI= Pkk3I21zdWIobWkoIlgiKSxtbigiMSIpKUc2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJJXRlbXBHRiQ3JC9JJF9DMUdGJCZJImNHRiQ2IyIiIi9JJF9DMkdGJCZGLjYjIiIj

LCYqJiZJImNHNiI2IyIiIkYoLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMqJkknbGFtYmRhR0YmI0YoIiIjSSJ4R0YmRihGKEYoKiYmRiU2I0YyRigtRio2IywkRi8hIiJGKEYo

Para essa solu\303\247\303\243o geral, aplicamos as condi\303\247\303\265es de fronteira, obtendo as equa\303\247\303\265es alg\303\251bricas homog\303\252neas

PiZJI0VxRzYiNiMiIiIvLUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRis2JEk3I21zdWIobWkoIlgiKSxtbigiMSIpKUdGJUkieEdGJS9GMSIiIUYz

LywmKiYmSSJjRzYiNiMiIiJGKUknbGFtYmRhR0YnI0YpIiIjRikqJiZGJjYjRixGKUYqRishIiIiIiE=

PiZJI0VxRzYiNiMiIiMvLUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRis2JEk3I21zdWIobWkoIlgiKSxtbigiMSIpKUdGJUkieEdGJS9GMUkjUGlHRisiIiE=

LywmKigmSSJjRzYiNiMiIiJGKUknbGFtYmRhR0YnI0YpIiIjLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiMqJkYqRitJI1BpR0YwRilGKUYpKigmRiY2I0YsRilGKkYrLUYuNiMsJEYzISIiRilGOyIiIQ==

O comando solve do Maple n\303\243o pode determinar valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para os quais esse sistema possui solu\303\247\303\265es n\303\243o-triviais. Guiamos o Maple a escrever a matriz do sistema como

Ji1JL0dlbmVyYXRlTWF0cml4RzYiNiQ3JCZJI0VxR0YlNiMiIiImRik2IyIiIzckJkkiY0dGJUYqJkYxRi1GKg==

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKilvMD0mKg==

e estabelecemos seu determinante para zero, a condi\303\247\303\243o sob a qual o sistema possir\303\241 solu\303\247\303\265es n\303\243o-triviais para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==.

Ly1JLERldGVybWluYW50RzYiNiMtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2Iy9JJCVpZEdGJSIqKW8wPSYqIiIh

LywmKiZJJ2xhbWJkYUc2IiIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCQqJkYlI0YnIiIjSSNQaUdGK0YnISIiRidGMyomRiVGJy1GKTYjRi9GJ0YnIiIh

Claramente, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjBGJ0YyRjI= \303\251 uma solu\303\247\303\243o, mas pertence ao primeiro caso. "Cancelando" LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy e solucionando, d\303\241

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1JIipHRiU2JC8sJiomSSdsYW1iZGFHRiciIiItSSRleHBHRiQ2IywkKiZGLyNGMCIiI0kjUGlHRiVGMCEiIkYwRjkqJkYvRjAtRjI2I0Y1RjBGMCIiISokRi9GOUYv

LCQqJEklX1ozfGlyRzYiIiIjISIi

onde novamente, substitu\303\255mos essa nota\303\247\303\243o por

QyQ+SSV0ZW1wRzYiJi1JJ2luZGV0c0clKnByb3RlY3RlZEc2IywkKiRJJV9aM3xpckdGJSIiIyEiIjYjIiIiRi8= LUklZXZhbEclKnByb3RlY3RlZEc2JCwkKiRJJV9aM3xpckc2IiIiIyEiIi9JJXRlbXBHRilJIm5HRik=

LCQqJEkibkc2IiIiIyEiIg==

de modo que os autovalores s\303\243o mais facilmente vistas como 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

Tendo determinado os autovalores, retornamos \303\240s equa\303\247\303\265es alg\303\251bricas determinamos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, e fazendo as substitui\303\247\303\265es \303\263bvias de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy para obter as equa\303\247\303\265es

QyU+JkkjRXFHNiI2IyIiJC1JKWFzc3VtaW5nR0koX3N5c2xpYkdGJjYkNyMtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0YrNiMtSSVldmFsR0YxNiQmRiU2IyIiIi9JJ2xhbWJkYUdGJiwkKiRJIm5HRiYiIiMhIiI3IydGPUkncG9zaW50R0YxRjg+JkYlNiMiIiUtRio2JDcjLUYvNiMtRjQ2JCZGJTYjRj5GOUZA

LywmKiheIyIiIkYmJkkiY0c2IjYjRiZGJkkibkdGKUYmRiYqKF4jISIiRiYmRig2IyIiI0YmRitGJkYmIiIh

LyoqXiMiIiJGJSkhIiJJIm5HNiJGJUYoRiUsJiZJImNHRik2I0YlRiUmRiw2IyIiI0YnRiUiIiE=

A solu\303\247\303\243o dessas equa\303\247\303\265es \303\251 agora

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JDwkJkkjRXFHRic2IyIiJCZGKzYjIiIlPCQmSSJjR0YnNiMiIiImRjM2IyIiIw==

PCQvJkkiY0c2IjYjIiIiJkYlNiMiIiMvRilGKQ==

e ent\303\243o as autofun\303\247\303\265es remanescentes s\303\243o

QyQ+SSV0ZW1wRzYiLUklc3Vic0clKnByb3RlY3RlZEc2Ji8mSSJjR0YlNiMiIiJJIkNHRiUvJkYsNiMiIiNGLy9JJ2xhbWJkYUdGJSwkKiRJIm5HRiVGMyEiIkk3I21zdWIobWkoIlgiKSxtbigiMSIpKUdGJUY5 LUkpYXNzdW1pbmdHSShfc3lzbGliRzYiNiQ3Iy1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHRiQ2Iy1JJmV2YWxjR0YqNiNJJXRlbXBHRiU3IydJIm5HRiVJJ3Bvc2ludEdGKw==

LCQqJkkiQ0c2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJIm5HRiVGJkkieEdGJUYmRiYiIiM=

Claramente, a constante LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= pode ser tida como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYkUSIyRidGMkYyLyUubGluZXRoaWNrbmVzc0dGMS8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Y+LyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjI=, ou como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkmbXNxcnRHRiQ2Iy1GIzYkLUkmbWZyYWNHRiQ2KC1GLDYmUScmIzk2MDtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYlUSIyRidGP0ZCRkIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlAvJSliZXZlbGxlZEdGPkZCRkJGK0ZC para obter autofun\303\247\303\265es ortonormais.

<Text-field style="Heading 1" layout="Heading 1">Exemplo: Uma EDO que n\303\243o \303\251 Auto-Adjunta</Text-field>

Conclu\303\255mos este artigo com um exemplo de um problema de autovalor no qual a equa\303\247\303\243o diferencial n\303\243o \303\251 auto-adjunta. Tal exemplo n\303\243o \303\251 coberto pela teoria de Sturm-Liouville, mas ilustra como o Maple pode ser guiado atrav\303\251s de alguns c\303\241lculos mais dif\303\255ceis que surgem em problemas de autovalores. Em particular, note as condi\303\247\303\265es de fronteira separadas de forma mais geral (mas ainda homog\303\252nea) no problema a seguir.

PkkkZGVxRzYiLywoLUklZGlmZkclKnByb3RlY3RlZEc2JS1JInlHRiQ2I0kieEdGJEYuRi4iIiItRig2JEYrRi4iIiMqJkknbGFtYmRhR0YkRi9GK0YvRi8iIiE= PiZJI2JjRzYiNiMiIiIvLCYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ5R0YlNiMiIiEiIiQtRjFGMiIiJkYz PiZJI2JjRzYiNiMiIiMvLCYtLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ5R0YlNiNJI1BpR0YuRictRjFGMiIiIiIiIQ==

LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ4R0YsRi5GLiIiIkYoIiIjKiZJJ2xhbWJkYUdGLEYvRipGL0YvIiIh

LywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieUdGKjYjIiIhIiIkLUYsRi0iIiZGLg==

LywmLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieUdGKjYjSSNQaUdGKCIiIy1GLEYtIiIiIiIh

Para obter solu\303\247\303\265es da equa\303\247\303\243o diferencial, procuramos um conjunto fundamental de membros independentes. Fazendo a escolha exponencial leva \303\240 equa\303\247\303\243o caracter\303\255stica

KiZJI0VHRzYiISIiLUkoY29sbGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUklZXZhbEdGKTYkSSRkZXFHRiQvLUkieUdGJDYjSSJ4R0YkRiNJJGV4cEdGKCIiIg==

LywoKiRJIm1HNiIiIiMiIiJGJUYnSSdsYW1iZGFHRiZGKCIiIQ==

cujas solu\303\247\303\265es s\303\243o

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC8sKCokSSJtR0YnIiIjIiIiRixGLUknbGFtYmRhR0YnRi4iIiFGLA==

NiQsJiEiIiIiIiokLCZGJUYlSSdsYW1iZGFHNiJGJCNGJSIiI0YlLCZGJEYlRiZGJA==

A partir dessas ra\303\255zes caracter\303\255sticas, podemos ver que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= \303\251 o caso de raiz repetida para o qual a solu\303\247\303\243o da EDO assume a forma especial

QyQ+SSV0ZW1wRzYiLUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQtSSVldmFsR0YpNiRJJGRlcUdGJS9JJ2xhbWJkYUdGJSIiIi1JInlHRiU2I0kieEdGJSEiIg== QyQ+STcjbXN1YihtaSgiWSIpLG1uKCIxIikpRzYiLUkodW5hcHBseUdGJTYkLUklZXZhbEclKnByb3RlY3RlZEc2JC1JJHJoc0dGKzYjSSV0ZW1wR0YlNyQvSSRfQzFHRiUmSSJyR0YlNiMiIiIvSSRfQzJHRiUmSSJzR0YlRjZJInhHRiUhIiI= Ly0uSTcjbXN1YihtaSgiWSIpLG1uKCIxIikpRzYiNiNJInhHRiYtRiVGJw==

Ly1JNyNtc3ViKG1pKCJZIiksbW4oIjEiKSlHNiI2I0kieEdGJSwmKiYmSSJyR0YlNiMiIiJGLS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRGJyEiIkYtRi0qKCZJInNHRiVGLEYtRi5GLUYnRi1GLQ==

Para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyLUY2Ni1RIixGJ0YyRjkvRjxRJXRydWVGJ0Y9Rj9GQUZDRkUvRkhRJjAuMGVtRicvRktRLDAuMzMzMzMzM2VtRidGMg==a solu\303\247\303\243o geral da EDO pode ser escrita como

QyQ+SSV0ZW1wRzYiLUknZHNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiRJJGRlcUdGJS1JInlHRiU2I0kieEdGJSEiIg== QyQ+STcjbXN1YihtaSgiWSIpLG1uKCIyIikpRzYiLUkodW5hcHBseUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklZXZhbEdGKTYkLUkkcmhzR0YpNiNJJXRlbXBHRiU3JC9JJF9DMUdGJSZJInJHRiU2IyIiIy9JJF9DMkdGJSZJInNHRiVGOEkieEdGJSEiIg== Ly0uSTcjbXN1YihtaSgiWSIpLG1uKCIyIikpRzYiNiNJInhHRiYtRiVGJw==

Ly1JNyNtc3ViKG1pKCJZIiksbW4oIjIiKSlHNiI2I0kieEdGJSwmKiYmSSJyR0YlNiMiIiMiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomLCYhIiJGLiokLCZGLkYuSSdsYW1iZGFHRiVGNyNGLkYtRi5GLkYnRi5GLkYuKiYmSSJzR0YlRixGLi1GMDYjKiYsJkY3Ri5GOEY3Ri5GJ0YuRi5GLg==

Embora o Maple padronizar para uma forma exponencial os membros de seu conjunto exponencial, n\303\243o dever\303\255amos assumir que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy \303\251 caracterizado por qualquer coisa diferente da condi\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIzEuRidGMkYy De fato, nesse ponto n\303\263s nem sabemos se os autovalores s\303\243o reais. Consequentemente, evitamos separar o c\303\241lculo nos casos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= porque poderiam implicar que sab\303\255amos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy n\303\243o era um n\303\272mero complexo.

Deixamos as condi\303\247\303\265es de limite determinar LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy considerando somente os casos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKyZOb3RFcXVhbDtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIzEuRidGMkYy No primeiro caso, obtemos as equa\303\247\303\265es alg\303\251bricas

PyhJImtHNiIiIiJGJSIiI0kldHJ1ZUclKnByb3RlY3RlZEc+JkkicUdGJDYjRiMtSSVldmFsR0YoNiQmSSNiY0dGJEYsL0kieUdGJEk3I21zdWIobWkoIlkiKSxtbigiMSIpKUdGJA==

LywmJkkickc2IjYjIiIiIiIjJkkic0dGJkYnIiIkIiIh

LywoKiYmSSJyRzYiNiMiIiJGKS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJzYjLCRJI1BpR0YtISIiRilGMiooJkkic0dGJ0YoRilGKkYpRjFGKUYyKiZGNEYpRipGKSIiIyIiIQ==

e no segundo,

PyhJImtHNiIiIiJGJSIiI0kldHJ1ZUclKnByb3RlY3RlZEc+JkkicUdGJDYjLCZGJkYlRiNGJS1JJWV2YWxHRig2JCZJI2JjR0YkNiNGIy9JInlHRiRJNyNtc3ViKG1pKCJZIiksbW4oIjIiKSlHRiQ=

LywqKiYmSSJyRzYiNiMiIiMiIiIsJiEiIkYqKiQsJkYqRipJJ2xhbWJkYUdGJ0YsI0YqRilGKkYqIiIkKiYmSSJzR0YnRihGKiwmRixGKkYtRixGKkYxRiUiIiZGM0Y2IiIh

LywqKigmSSJyRzYiNiMiIiMiIiIsJiEiIkYqKiQsJkYqRipJJ2xhbWJkYUdGJ0YsI0YqRilGKkYqLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YnNiMqJkYrRipJI1BpR0Y0RipGKkYpKigmSSJzR0YnRihGKiwmRixGKkYtRixGKi1GMjYjKiZGPEYqRjhGKkYqRikqJkYlRipGMUYqRioqJkY6RipGPUYqRioiIiE=

Os determinantes do sistema de matrizes em cada caso s\303\243o ent\303\243o

PiZJIlFHNiI2IyIiIi1JLERldGVybWluYW50R0YlNiMmLUkvR2VuZXJhdGVNYXRyaXhHRiU2JDckJkkicUdGJUYmJkYxNiMiIiM3JCZJInJHRiVGJiZJInNHRiVGJkYm

LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJI1BpR0YnISIiIiIiRixGLiEiI0YkIiIo

PiZJIlFHNiI2IyIiIy1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JC1JLERldGVybWluYW50R0YlNiMmLUkvR2VuZXJhdGVNYXRyaXhHRiU2JDckJkkicUdGJTYjIiIkJkY3NiMiIiU3JCZJInJHRiVGJiZJInNHRiVGJjYjIiIiNyNJJGV4cEdGKg==

LCYqJiwoIiIpIiIiKiQsJkYmRiZJJ2xhbWJkYUc2IiEiIiNGJiIiIyEiKEYpISInRiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IyomLCZGK0YmRidGJkYmSSNQaUdGM0YmRiZGJiomLCghIilGJkYnRi5GKSIiJ0YmLUYxNiMsJComLCZGJkYmRidGJkYmRjhGJkYrRiZGJg==

respectivamente. Desde que o determinante em seja n\303\243o-zero, devemos ter 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ent\303\243o a solu\303\247\303\243o correspondente \303\251 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= n\303\243o \303\251 portanto, um autovalor.

Existem zeros do determinante em , alguns dos quais s\303\243o

LUkpQW5hbHl0aWNHNiI2JSZJIlFHRiQ2IyIiIy9JI3JlR0YkOyEjNSIjNS9JI2ltR0YkRiw=

NickIishPXImZVghIzUkIiskZWNqNiIhIiokIiszdFNWVUYoJCIrJilcZF4jKkYoJCIrZysrKzVGKA==

O comando Analytic no pacote RootFinding acha todas as ra\303\255zes reais e complexasem uma regi\303\243o do plano complexo. N\303\263s j\303\241 sabemos que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI= n\303\243o \303\251 um autovalor, ent\303\243o temos os autovalores reais

LUkncmVtb3ZlRyUqcHJvdGVjdGVkRzYkZio2I0kid0c2IkYpNiRJKW9wZXJhdG9yR0YpSSZhcnJvd0dGKUYpLUkjaXNHNiRGJEkoX3N5c2xpYkdGKTYjMi1JJGFic0dGJDYjLCY5JCIiIiEiIkY4JEY4ISIkRilGKUYpNyckIishPXImZVghIzUkIiskZWNqNiIhIiokIiszdFNWVUZCJCIrJilcZF4jKkZCJCIrZysrKzVGQg==

NyYkIishPXImZVghIzUkIiskZWNqNiIhIiokIiszdFNWVUYoJCIrJilcZF4jKkYo

Enquanto o comando Analytic nos d\303\241 uma garantia que todas as solu\303\247\303\265es em uma regi\303\243o fixa podem ser achadas, ele \303\251 devagar. Se aceitarmos como evid\303\252ncia que os autovalores s\303\243o positivos e reais, ent\303\243o podemos encontr\303\241-los um pouco mais r\303\241pido mudando a forma do determinante em para

LUkpYXNzdW1pbmdHSShfc3lzbGliRzYiNiQ3Iy1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHRiQ2Iy1JJmV2YWxjR0YqNiMmSSJRR0YlNiMiIiM3Iy1JIj5HRis2JEknbGFtYmRhR0YlIiIi

KiheIyEiIyIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJI1BpR0YpISIiRiUsKComLCZGL0YlSSdsYW1iZGFHRitGJSNGJSIiIy1JJGNvc0dGKDYjKiZGMkY0Ri5GJUYlIiIoLUkkc2luR0YoRjghIikqJkY7RiVGM0YlIiInRiU=

e usando o comando Roots para encontrar

PkkiTEc2Ii1JJlJvb3RzR0YkNiUqKl4jISIjIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMsJEkjUGlHRi8hIiJGKywoKiYsJkY0RitJJ2xhbWJkYUdGJEYrI0YrIiIjLUkkY29zR0YuNiMqJkY3RjlGM0YrRisiIigtSSRzaW5HRi5GPSEiKSomRkBGK0Y4RisiIidGKy1JIi9HRi42IyomXiNGK0YrRixGK0YrL0Y4OyRGK0Y0IiNFL0kobnVtZXJpY0dGL0kldHJ1ZUdGLw==

NygkIisiPXImZVghIzUkIiskZWNqNiIhIiokIis0dFNWVUYoJCIrKClcZF4jKkYoJCIrXFxURDshIikkIisoekhiXyNGLw==

Para qualquer tal autovalor, as equa\303\247\303\265es LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== s\303\243o redundantes, ent\303\243o podemos solucionar um deles para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, digamos, em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, obtendo

PkkiU0c2Ii1JJnNvbHZlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQmSSJxR0YkNiMiIiQmSSJzR0YkNiMiIiM=

KigmSSJyRzYiNiMiIiMiIiIsJkYnRigqJCwmRihGKEknbGFtYmRhR0YlISIiI0YoRiciIiRGKCwmISIjRihGKkYvRi0=

Para qualquer autovalor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, a autofun\303\247\303\243o correspondente \303\251 ent\303\243o dada (para uma constante multiplicativa) por

PkkiWUc2IiomJkkickdGJDYjIiIjISIiLUkoY29sbGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUklZXZhbEdGLjYkLUk3I21zdWIobWkoIlkiKSxtbigiMiIpKUdGJDYjSSJ4R0YkLyZJInNHRiRGKEkiU0dGJDckRiZJJGV4cEdGLSIiIg==

LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomLCYhIiIiIiIqJCwmRi1GLUknbGFtYmRhR0YoRiwjRi0iIiNGLUYtSSJ4R0YoRi1GLSooLCZGMkYtRi4iIiRGLSwmISIjRi1GLkY2RiwtRiQ2IyomLCZGLEYtRi5GLEYtRjNGLUYtRi0=

Com LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI=, os termos exponenciais ter\303\243o equivalentes trigonom\303\251tricos, ent\303\243o escrevemos

LUkpYXNzdW1pbmdHSShfc3lzbGliRzYiNiQ3Iy1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHRiQ2Iy1JJmV2YWxjR0YqNiNJIllHRiU3Iy1JIj5HRis2JEknbGFtYmRhR0YlIiIi

LCQqKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikhIiIiIiIsLC1JJGNvc0dGJjYjKiYsJkYtRi5JJ2xhbWJkYUdGKUYuI0YuIiIjRixGLiIiKiomRjBGLkY1Ri4hIioqJkY0RjYtSSRzaW5HRiZGMkYuIiInKiZeIyEiJUYuRjxGLkYuKiheI0Y+Ri5GNEY2RjBGLkYuRi4sJiEiJkYuRjVGOEYtISIj

e ent\303\243o

LUkoY29sbGVjdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLCQqKC1JJGV4cEdGJDYjLCRJInhHRichIiIiIiIsLC1JJGNvc0dGJDYjKiYsJkYwRjFJJ2xhbWJkYUdGJ0YxI0YxIiIjRi9GMSIiKiomRjNGMUY4RjEhIioqJkY3RjktSSRzaW5HRiRGNUYxIiInKiZeIyEiJUYxRj9GMUYxKiheI0ZBRjFGN0Y5RjNGMUYxRjEsJiEiJkYxRjhGO0YwISIjNyRGQEY0

LCYqKi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikhIiIiIiIsJiokLCZGLUYuSSdsYW1iZGFHRilGLiNGLiIiIyIiJ14jISIlRi5GLiwmISImRi5GMiIiKkYtLUkkc2luR0YmNiMqJkYxRjNGLEYuRi4hIiMqKkYkRi4sKEY6Ri5GMiEiKiomXiNGNUYuRjFGM0YuRi5GOEYtLUkkY29zR0YmRj1GLkY/

As partes real e imagin\303\241ria s\303\243o respectivamente

QyQ+SSNZYUc2Ii1JKWFzc3VtaW5nR0koX3N5c2xpYkdGJTYkNyMtSSZldmFsY0c2JCUqcHJvdGVjdGVkR0YoNiMtSSNSZUdGLjYjLCYqKi1JJGV4cEdGLTYjLCRJInhHRiUhIiIiIiIsJiokLCZGOkY7SSdsYW1iZGFHRiVGOyNGOyIiIyIiJ14jISIlRjtGOywmISImRjtGPyIiKkY6LUkkc2luR0YtNiMqJkY+RkBGOUY7RjshIiMqKkY1RjssKEZHRjtGPyEiKiomXiNGQkY7Rj5GQEY7RjtGRUY6LUkkY29zR0YtRkpGO0ZMNyMtSSI+R0YuNiRGP0Y7Rjs= PkkjWWJHNiItSSlhc3N1bWluZ0dJKF9zeXNsaWJHRiQ2JDcjLUkmZXZhbGNHNiQlKnByb3RlY3RlZEdGJzYjLUkjSW1HRi02IywmKiotSSRleHBHRiw2IywkSSJ4R0YkISIiIiIiLCYqJCwmRjlGOkknbGFtYmRhR0YkRjojRjoiIiMiIideIyEiJUY6RjosJiEiJkY6Rj4iIipGOS1JJHNpbkdGLDYjKiZGPUY/RjhGOkY6ISIjKipGNEY6LChGRkY6Rj4hIioqJl4jRkFGOkY9Rj9GOkY6RkRGOS1JJGNvc0dGLEZJRjpGSzcjLUkiPkdGLTYkRj5GOg==

LCYqKi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikhIiIiIiIsJkYtRi5JJ2xhbWJkYUdGKUYuI0YuIiIjLCYhIiZGLkYwIiIqRi0tSSRzaW5HRiY2IyomRi9GMUYsRi5GLiEjNyoqRiRGLiwmRjVGLkYwISIqRi5GM0YtLUkkY29zR0YmRjhGLiEiIw==

LCYqKC1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikhIiIiIiIsJiEiJkYuSSdsYW1iZGFHRikiIipGLS1JJHNpbkdGJjYjKiYsJkYtRi5GMUYuI0YuIiIjRixGLkYuIiIpKipGJEYuRjdGOEYvRi0tSSRjb3NHRiZGNUYuISM3

De fato, cada parte real e imagin\303\241ria \303\251 separadamente uma solu\303\247\303\243o. Mas elas n\303\243o podem ser solu\303\247\303\265es independentes - elas devem ser m\303\272ltiplas uma da outra. Esse m\303\272ltiplo comum \303\251

PkkiZkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMtSShjb252ZXJ0R0YoNiQtSSd0YXlsb3JHRig2JSomSSNZYUdGJCIiIkkjWWJHRiQhIiJJInhHRiQiIiRJKHBvbHlub21HRig=

LCQqJCwmISIiIiIiSSdsYW1iZGFHNiJGJiNGJiIiIyMhIiRGKg==

e n\303\263s podemos verificar que, de fato, as partes real e imagin\303\241ria de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== s\303\243o linearmente dependentes mostrando que a combina\303\247\303\243o linear 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 desaparece.

LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmKiZJI1liR0YnIiIiSSJmR0YnRixGLEkjWWFHRichIiI=

IiIh

O m\303\272ltiplo comum pode ser encontrado com

PkkiZ0c2Ii1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJI1lhR0YkL0kieEdGJCIiIQ==

LCQqJiwmIiIqIiIiSSdsYW1iZGFHNiIhIipGJiwmISImRiZGJ0YlISIiISIj

e uma forma simplificada da parte real \303\251 ent\303\243o

PkkjWWNHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQqJkkjWWFHRiQiIiJJImdHRiQhIiI3JUkkZXhwR0YnSSRjb3NHRidJJHNpbkdGJw==

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiYsJiEiIiIiIkknbGFtYmRhR0YpRi4jRi4iIiNJInhHRilGLkYuKihGLEYwLUkkc2luR0YmRipGLiwmIiIqRi5GLyEiKkYtIiInRi4tSSRleHBHRiY2IywkRjJGLUYu

Usando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==, podemos obter as primeiras seis autofun\303\247\303\265es como

LUkpdW5hc3NpZ25HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy5JJHBoaUdGJw== QyQ+JkkkcGhpRzYiNiMiIiItSSlhc3N1bWluZ0dJKF9zeXNsaWJHRiY2JDcjLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdGKzYjLUknZXhwYW5kR0YxNiMtSShjb252ZXJ0R0YxNiQtSSVldmFsR0YxNiRJI1ljR0YmL0knbGFtYmRhR0YmJkkiTEdGJkYnSSRleHBHRjA3Iy1JIj5HRjE2JEkieEdGJiIiIUYo PyhJImtHNiIiIiMiIiItSSVub3BzRyUqcHJvdGVjdGVkRzYjSSJMR0YkSSV0cnVlR0YpPiZJJHBoaUdGJDYjRiMtSSVldmFsR0YpNiRJI1ljR0YkL0knbGFtYmRhR0YkJkYrRjA=

LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ4R0YoJCErISplUkJFISM1JCIqXykzN1tGLi1GJDYjLCRGKyQhKzYvbVA8ISIqJCIrWzZ6PSYqRi4=

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikkIit2ITM2VCQhIzUiIiItSSRzaW5HRiZGKiQhK0ciKlJhPiEiKkYwLUkkZXhwR0YmNiMsJEYsISIiRjA=

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikkIitFaSU0IT0hIioiIiItSSRzaW5HRiZGKiQhK2d4dixQISM1RjAtSSRleHBHRiY2IywkRiwhIiJGMA==

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikkIishW2JEKEchIioiIiItSSRzaW5HRiZGKiQhKyQqUSIzSyMhIzVGMC1JJGV4cEdGJjYjLCRGLCEiIkYw

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikkIishNGNjIVIhIioiIiItSSRzaW5HRiZGKiQhKyE+RXBxIiEjNUYwLUkkZXhwR0YmNiMsJEYsISIiRjA=

KiYsJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInhHRikkIisqem1cI1whIioiIiItSSRzaW5HRiZGKiQhK0Zxa2A4ISM1RjAtSSRleHBHRiY2IywkRiwhIiJGMA==

Um gr\303\241fico das seis primeiras autofun\303\247\303\265es \303\251 dado na figura 1, onde nossa normaliza\303\247\303\243o possui todas as autofun\303\247\303\265es passando por (0, 1). As autofun\303\247\303\265es 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\303\243o dadas em preto pontilhado, preto s\303\263lido, azul, vermelho, verde, e magenta.

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyMtSSRzZXFHRiU2JCZJJHBoaUdGJzYjSSJrR0YnL0YwOyIiIiIiJy9JInhHRic7IiIhSSNQaUdGJS9JJmNvbG9yR0YnNyhJJmJsYWNrR0YnRj1JJWJsdWVHRidJJHJlZEdGJ0kmZ3JlZW5HRidJKG1hZ2VudGFHRicvSSpsaW5lc3R5bGVHRic3JCIiIy1JIiRHRiU2JEYzIiIm

-%%PLOTG6,-%'CURVESG6%7dw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"0w.qN+'G(*!#:7$$"0p4!4F)))3$!#;$"0&H-%Rr')\*!#:7$$"0FS')[;^q%!#;$"0"G3G^"oC*!#:7$$"0*y*[rj?L'!#;$"0#ei0bG+!*!#:7$$"06')\My7&z!#;$"0@XM^.<w)!#:7$$"08y4e(\_%*!#;$"0_*)yk#QY&)!#:7$$"0h$e9Bp+6!#:$"09&>%[G#H$)!#:7$$"/T')G7Xh7!#9$"0QsKN(o5")!#:7$$"0onse%p@9!#:$"0LSz(Hz)*y!#:7$$"0B)zxX_'e"!#:$"/74kb&oo(!#97$$"0OwJ#pqJ<!#:$"0AV`h.^](!#:7$$"0<Fx&y9&*=!#:$"0e,-"='eI(!#:7$$"0Jek*)f#f?!#:$"0&3I*4o86(!#:7$$"02(*\D7u@#!#:$"0CNXgo!Hp!#:7$$"0)p"yuH5O#!#:$"02X!p/xnn!#:7$$"0;hx$[!=`#!#:$"0ey%374"e'!#:7$$"0J\&RPZwE!#:$"0AJ&Hq=Fk!#:7$$"0$>Lk!eZ%G!#:$"/Z33a"HD'!#97$$"0.Yk?=P*H!#:$"0W3n<5G5'!#:7$$"0od74^r:$!#:$"0ar1^'[Uf!#:7$$"0%*3h%yx7L!#:$"0v.:'G&Rz&!#:7$$"0cx%Q!e^Z$!#:$"0/$)>2`Jk&!#:7$$".#oWPFCO!#8$"0W6)z5L3b!#:7$$"0nb'y^6&y$!#:$"/VCh!HnO&!#97$$"0"G*[A%=_R!#:$"0/81vHPA&!#:7$$"0%=x?$=w4%!#:$"0H(z\ha-^!#:7$$"0X='G6paU!#:$"04g)y)>](\!#:7$$"/$4(3F'pT%!#9$"0Ce@4No%[!#:7$$"0kf;z8dd%!#:$"0,6#Q"R[s%!#:7$$"/FCFOJHZ!#9$"0Q$**QJ#*4Y!#:7$$"0i-m0g)**[!#:$"00O-8Ee[%!#:7$$"0RT(3O5`]!#:$"0nXwIttP%!#:7$$"/3$)4Qs;_!#9$"0x#p7!zYE%!#:7$$"/8wI!))\O&!#9$"0<8lua_;%!#:7$$"0(4&f)*zq_&!#:$"0WBy')3%fS!#:7$$"0*Hm$)efzc!#:$"0#pf#yjC'R!#:7$$"0VIA%*=!Re!#:$"0,_R9)zjQ!#:7$$"0@i=7%)[*f!#:$"0-C:V-*pP!#:7$$"0j.,_g!eh!#:$"0M][e[Un$!#:7$$"04&e&y=_J'!#:$"0T.N57Ye$!#:7$$"0664"e$fZ'!#:$"0&R'y'HS&\$!#:7$$"0;B"=?KNm!#:$"07lsj>$4M!#:7$$"0"*R$*Q"y"y'!#:$"0e'*=0eAL$!#:7$$"0(eoc.k\p!#:$"0z2l#[FYK!#:7$$"0PT"=ax*4(!#:$"/KxK+VrJ!#97$$"0"Hl22&)fs!#:$"0mS&ePr$4$!#:7$$"0<i[=jIT(!#:$"/">)*\!H@I!#97$$"0c>2&f=$e(!#:$"0B^3LfI%H!#:7$$"0)[nl(*\Ix!#:$"0)=@j]7xG!#:7$$"0p%HjAq(*y!#:$"0%R3ujE/G!#:7$$"0$pO)=*3]!)!#:$"0LA%RgkRF!#:7$$"06Le`wn@)!#:$"025EBa3n#!#:7$$"0#4qE&*fg$)!#:$"0i&yce/8E!#:7$$"0Cg^7(RC&)!#:$"0-XN_2*[D!#:7$$"0^8*=vh#o)!#:$"04%f6^i)[#!#:7$$"0>66bM2%))!#:$"0OxApy*HC!#:7$$"0ru#**)p#)**)!#:$"0@b1\#4tB!#:7$$"0$=h]?h\"*!#:$"0OHF&f&)>B!#:7$$"0A-U#p@8$*!#:$"0'G#ox@QE#!#:7$$"0O"fJU9p%*!#:$"07YEoV=@#!#:7$$"0(4B!Q]Kj*!#:$"0AYg_,'e@!#:7$$"0[nD'f!=y*!#:$"0AeRLj;6#!#:7$$"0-!)4p()f%**!#:$"0"o=9;9h?!#:7$$"0Q6x^C.,"!#9$"0hL]sZS,#!#:7$$"0\7%*)>,E5!#9$"0`maf)Ho>!#:7$$"0n1Wh(RU5!#9$"0)3yb'*y@>!#:7$$"0"[?t$*[d5!#9$"0h1(oe2!)=!#:7$$"0pl&)RXH2"!#9$"0by*Q7WQ=!#:7$$"0W!3(o5+4"!#9$"0Fe*QOr$z"!#:7$$"0t9!3>Y06!#9$"0eY$Q,Ja<!#:7$$"0`#fvOE@6!#9$"0F-^<e]r"!#:7$$"0%R>mOLP6!#9$"0JpV1&>w;!#:7$$"02yY60@:"!#9$"/-`AfQT;!#97$$"0tE/+^y;"!#9$"0(=4N'=_g"!#:7$$"0apu-xM="!#9$"0@Q=he-d"!#:7$$"0OZ#>?>+7!#9$"/@d!)='Q`"!#97$$"0'**RvA%\@"!#9$"0J"G9'yD]"!#:7$$"0v%)*35'>B"!#9$"0&Q5i/Vn9!#:7$$"0[cR5>tC"!#9$"0[J'=*elV"!#:7$$"0Thh8:DE"!#9$"/A!yg#y19!#97$$"0E_Dvb)y7!#9$"0T5yz%fv8!#:7$$"0y/,Zi_H"!#9$"0#f&p$R7X8!#:7$$"0F!y`,>58!#9$"0Dm[59"=8!#:7$$"0Ee2pUgK"!#9$"09@*ye:!H"!#:7$$"/)*[<K^T8!#8$"0^))>]sNE"!#:7$$"0/y;V$Ge8!#9$"0Z"p'y;bB"!#:7$$"0(3dHVzs8!#9$"/0puE'=@"!#97$$"0OZLig%*Q"!#9$"0R0&*e(Q&="!#:7$$"0Ctc"y709!#9$"0&oUKo:h6!#:7$$"012+oV1U"!#9$"/$yZzqx8"!#97$$"0(oK()=<O9!#9$"0U">v%e\6"!#:7$$"0Dpen4?X"!#9$"0)>#f_&G#4"!#:7$$"/,Smuzo9!#8$"0s^J")*))o5!#:7$$"0w*RE->%["!#9$"0=Yv'))*z/"!#:7$$"0"R/d[Q*\"!#9$"0i8%=())y-"!#:7$$"0jIrmFc^"!#9$"0HBxzQp+"!#:7$$"0`R!=^$=`"!#9$"0=='y<$e')*!#;7$$"0P,xE#QY:!#9$"02plFrwo*!#;7$$"00;bT9Oc"!#9$"0(H(GXP?[*!#;7$$"0Z4xZL!y:!#9$"0i+")QjVJ*!#;7$$"0/J0>)*[f"!#9$"0ul<L'=B"*!#;7$$"0MU^#R*4h"!#9$"0wkwNdb%*)!#;7$$"0`E*)\l`i"!#9$"0H@mIO3z)!#;7$$"0$GsOy_T;!#9$"0rk')\%4@')!#;7$$"0T&)*3tzd;!#9$"00$pU*fYX)!#;7$$"0H'Gb%*)Rn"!#9$"0P%[J0J$H)!#;7$$"0@Ywk,!*o"!#9$"0.J>IVu9)!#;7$$"0z%H.fa/<!#9$"0v)*4v#3+!)!#;7$$"0%Gs%z@1s"!#9$"0Hb)fK^^y!#;7$$"/KcIhkO<!#8$"0Ud*oE>2x!#;7$$"0D;'H"HJv"!#9$"0%oeBJciv!#;7$$"02aTOZww"!#9$"0;))z^)HQu!#;7$$"0:4wX"*Ry"!#9$"0DU?"\"=I(!#;7$$"0E#[hES+=!#9$"0-Sj#pIor!#;7$$"09Ot*y@;=!#9$"/G!333H/(!#:7$$"08=mkz0$=!#9$"0*Gl;RtJp!#;7$$"0b7c:dw%=!#9$"0Z#y3Hz-o!#;7$$"0O"zXS7i=!#9$"0[4[seip'!#;7$$"0jp#yC&*y=!#9$"0r$H#\g`d'!#;7$$"0/"[#\[Q*=!#9$"04CC-!*4Z'!#;7$$"/A'4y">5>!#8$"0&*=COf#fj!#;7$$"0LZkXad#>!#9$"0h$>l")\bi!#;7$$"0>%ovC*>%>!#9$"0["z]")*)\h!#;7$$"0j/j//p&>!#9$"0<ifmd_0'!#;7$$".-(*=))H(>!#7$"0$H&G+8c&f!#;7$$"0rDV4&p*)>!#9$"0Bm%>urae!#;7$$"0i8R]QU+#!#9$"0"4`')H**od!#;7$$"0G)p%yX*>?!#9$"0Bz`Y^&yc!#;7$$"0O2F%H<O?!#9$"0h4^)HR(e&!#;7$$"/C+^![?0#!#8$"0mj"y4R+b!#;7$$"/2cM!3u1#!#8$"0UkQk3#=a!#;7$$"0p'\xEY%3#!#9$"0'QXav>H`!#;7$$"0d5F.(y*4#!#9$"0D<n^s6D&!#;7$$"0^>G0\h6#!#9$"08<ub^)p^!#;7$$"0c7\Zv48#!#9$"0MVU:*)y4&!#;7$$"0`J/n%=Z@!#9$"0&3Z9P/@]!#;7$$"0t--EOC;#!#9$"0Ns'\qV]\!#;7$$"0[fgcy$y@!#9$"/Vb2pMy[!#:7$$"0lAS3lR>#!#9$"/bm(G:&4[!#:7$$"/o%Qs#G5A!#8$"0^S\$f:RZ!#;7$$"0%\R]&)*fA#!#9$"0[wws#*Hn%!#;7$$"0aFHDq?C#!#9$"0Vg<k7pg%!#;7$$"0v[O()3!eA!#9$"0Jc-,<Ha%!#;7$$"0Uq2"[lsA!#9$"037&*GEa[%!#;7$$"0-0vqS%*G#!#9$"0cm33V5U%!#;7$$"0d]O@aWI#!#9$"0Fbj'pykV!#;7$$"0s,Euh/K#!#9$"0"=5/=:1V!#;7$$"0lA.*HyNB!#9$"/RFwwH^U!#:7$$"0R3pE&z_B!#9$"0z<wZ-=>%!#;7$$"0#RS[m_nB!#9$"0$oR/TXTT!#;7$$"/f;)*oC%Q#!#8$"0**R*z()e&3%!#;7$$"07t1f&[*R#!#9$"0xH+$)Ge.%!#;7$$"0u>aKahT#!#9$"0TTk$pi#)R!#;7$$"0_1XiO0V#!#9$"0Y#=]rsPR!#;7$$"0Y_VQ;pW#!#9$"0C@V9&p()Q!#;7$$"0yFPUQFY#!#9$"0hed#RXSQ!#;7$$"0bZp7]&yC!#9$"08!)pl#G%z$!#;7$$"/RwhOI%\#!#8$"/%\))G(G\P!#:7$$"0i(*o(yV4D!#9$"0XB)[A)pq$!#;7$$"0lcUO)zDD!#9$"0m\NjQAm$!#;7$$"0d&*\4"RTD!#9$"/.#*=l_?O!#:7$$"0`f)4<!yb#!#9$"0T"p2EexN!#;7$$"0=$4oslsD!#9$"0Lh`'=`RN!#;7$$"0VM4Wv!*e#!#9$"0%)G:el$)\$!#;7$$"0"ya*=,[g#!#9$"0uH47'yfM!#;7$$"0#*[7m)[?E!#9$"0o[Q;3@U$!#;7$$"/JC'Guoj#!#8$"03EN:#f$Q$!#;7$$"0DT]/m>l#!#9$"0'>!G:c)[L!#;7$$"07-/2Aum#!#9$"0s8T2(*RJ$!#;7$$"0(o"*et[%o#!#9$"0F7FaFjF$!#;7$$"0;^)z&Q**p#!#9$"0P"*f'H%HC$!#;7$$"0(*GuMSdr#!#9$"0P*=@@\4K!#;7$$"0QI!Q."=t#!#9$"0Njr$=<wJ!#;7$$"0^9ly"eYF!#9$"04W"o%\h9$!#;7$$"0<jAnFBw#!#9$"-)G3pZ6$!#87$$"0(fI*p`zx#!#9$"0%og^nC%3$!#;7$$"0z$3"poYz#!#9$"0CdfRiA0$!#;7$$"/kBZ*=%4G!#8$"0_YS0%fCI!#;7$$"0=@3oPk#G!#9$"0d(HM=I$*H!#;7$$"0"HzvdzTG!#9$"/%3o))Gc'H!#:7$$"0%y*z!=*p&G!#9$"0hWz!GwQH!#;7$$"/()QCCLtG!#8$"0$Q()*=I/"H!#;7$$"0@T>9R(*)G!#9$"0v=\!)[D)G!#;7$$"/nhDom/H!#8$"0u<b"4mdG!#;7$$"0p%fi$>0#H!#9$"0>\G@><$G!#;7$$"0BE$*))*)f$H!#9$"0C)oSa(o!G!#;7$$"0[9N5gF&H!#9$"0)fmO&e/y#!#;7$$"/tS,5FnH!#8$"0CA&[8-eF!#;7$$"0z$=&HPR)H!#9$"0B&HB3sKF!#;7$$"0o4v[/'**H!#9$"0&e%=D$Q4F!#;7$$"0\V=N?^,$!#9$"083&4lo'o#!#;7$$"/L;f&[1.$!#8$"0z"Qw`PkE!#;7$$"0o0xM'[YI!#9$"0JPx6D?k#!#;7$$"0`O#QTFjI!#9$"/1()e,x=E!#:7$$"0>O#)*omyI!#9$"0'*oM%>$yf#!#;7$$"0M!))G:'Q4$!#9$"0"**yT[^xD!#;7$$"01n*QV55J!#9$"0W#y=I<cD!#;7$$"0'f()*y6j7$!#9$"0%zo,hDND!#;7$$"-X]EfTJ!#6$"0'oBBE(e^#!#;-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""#-%'CURVESG6%7dw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"/O3vtrF(*!#97$$"0p4!4F)))3$!#;$"/;&o7Pc\*!#97$$"0FS')[;^q%!#;$"0TbDV7*R#*!#:7$$"0*y*[rj?L'!#;$"/,i?(H!))*)!#97$$"06')\My7&z!#;$"/bA2lvU()!#97$$"08y4e(\_%*!#;$"0x)3c=4?&)!#:7$$"0h$e9Bp+6!#:$"01q'e&\UH)!#:7$$"/T')G7Xh7!#9$"0pk7yOc1)!#:7$$"0onse%p@9!#:$"04LU+xE%y!#:7$$"0B)zxX_'e"!#:$"0X'y%Gj$=w!#:7$$"0OwJ#pqJ<!#:$"00dFQL\U(!#:7$$"0<Fx&y9&*=!#:$"0#*\hDP<@(!#:7$$"0Jek*)f#f?!#:$"0p`&f`V-q!#:7$$"02(*\D7u@#!#:$"0&elH,<0o!#:7$$"0)p"yuH5O#!#:$"0*y([EP(Hm!#:7$$"0;hx$[!=`#!#:$"0;GRNKcU'!#:7$$"0J\&RPZwE!#:$"0(f=&e&[ci!#:7$$"0$>Lk!eZ%G!#:$"0k%>\-)R1'!#:7$$"0.Yk?=P*H!#:$"/N;*HEt*e!#97$$"0od74^r:$!#:$"0cO`8]%=d!#:7$$"0%*3h%yx7L!#:$"06$*RT1>b&!#:7$$"0cx%Q!e^Z$!#:$"0GF)G)**>Q&!#:7$$".#oWPFCO!#8$"0_l([NQH_!#:7$$"0nb'y^6&y$!#:$"0x3@)zNo]!#:7$$"0"G*[A%=_R!#:$"0o*H0*o\!\!#:7$$"0%=x?$=w4%!#:$"0s*=(e))ew%!#:7$$"0X='G6paU!#:$"0$enU#4*=Y!#:7$$"/$4(3F'pT%!#9$"0(zK?H_qW!#:7$$"0kf;z8dd%!#:$"/swh/pGV!#97$$"/FCFOJHZ!#9$"0)zW-!RX>%!#:7$$"0i-m0g)**[!#:$"0Ow!zW1\S!#:7$$"0RT(3O5`]!#:$"0QWC*)49#R!#:7$$"/3$)4Qs;_!#9$"0HgPvX#)y$!#:7$$"/8wI!))\O&!#9$"0a)3Z:KqO!#:7$$"0(4&f)*zq_&!#:$"0$*p9aCVa$!#:7$$"0*Hm$)efzc!#:$"0,'))3#3&GM!#:7$$"0VIA%*=!Re!#:$"0/"GF!G-J$!#:7$$"0@i=7%)[*f!#:$"0Oq_w#G(>$!#:7$$"0j.,_g!eh!#:$"0<Qz?Q=3$!#:7$$"04&e&y=_J'!#:$"0"e]TUItH!#:7$$"0664"e$fZ'!#:$".eI!3&\'G!#87$$"0;B"=?KNm!#:$"0u@z%o2gF!#:7$$"0"*R$*Q"y"y'!#:$"/hN![Jfm#!#97$$"0(eoc.k\p!#:$"0r8:d'fgD!#:7$$"0PT"=ax*4(!#:$"0$H%3kg'oC!#:7$$"0"Hl22&)fs!#:$"0"pH#HeHP#!#:7$$"0<i[=jIT(!#:$"0WbM8eNG#!#:7$$"0c>2&f=$e(!#:$"0$Q%)p%fn=#!#:7$$"0)[nl(*\Ix!#:$"/i.%y&*\5#!#97$$"0p%HjAq(*y!#:$"0;G"*oeW,#!#:7$$"0$pO)=*3]!)!#:$"00E'*>+S$>!#:7$$"06Le`wn@)!#:$"0)>!p?!>[=!#:7$$"0#4qE&*fg$)!#:$"0*)oK[gfx"!#:7$$"0Cg^7(RC&)!#:$"0.`&R'*p&p"!#:7$$"0^8*=vh#o)!#:$"/;O"y`,i"!#97$$"0>66bM2%))!#:$"0#o')))GcY:!#:7$$"0ru#**)p#)**)!#:$"0uXsQ)4v9!#:7$$"0$=h]?h\"*!#:$"0kU@\_"39!#:7$$"0A-U#p@8$*!#:$"0L)y@uiP8!#:7$$"0O"fJU9p%*!#:$"0emh_f@F"!#:7$$"0(4B!Q]Kj*!#:$"0j#\))3107!#:7$$"0[nD'f!=y*!#:$"04<D[#)e9"!#:7$$"0-!)4p()f%**!#:$"0tD%Q)p@3"!#:7$$"0Q6x^C.,"!#9$"0;LU5uF-"!#:7$$"0\7%*)>,E5!#9$"/GM]Ky]'*!#:7$$"0n1Wh(RU5!#9$"0fzF&[Qk!*!#;7$$"0"[?t$*[d5!#9$"0&p8F&f'Q&)!#;7$$"0pl&)RXH2"!#9$"0em$*fFU,)!#;7$$"0W!3(o5+4"!#9$"/Viv!z7X(!#:7$$"0t9!3>Y06!#9$"0s4(*[De&p!#;7$$"0`#fvOE@6!#9$"0f(G%oMGY'!#;7$$"0%R>mOLP6!#9$"0%>djYQvf!#;7$$"02yY60@:"!#9$"061WDM%Rb!#;7$$"0tE/+^y;"!#9$"0php!*Hs3&!#;7$$"0apu-xM="!#9$"0UUrJd4l%!#;7$$"0OZ#>?>+7!#9$"/([:D_x>%!#:7$$"0'**RvA%\@"!#9$"08uN<N"4Q!#;7$$"0v%)*35'>B"!#9$"/t')4@ktL!#:7$$"0[cR5>tC"!#9$"0$*y2[TA*H!#;7$$"0Thh8:DE"!#9$"/)Hwsdai#!#:7$$"0E_Dvb)y7!#9$"0(\xn'[DC#!#;7$$"0y/,Zi_H"!#9$"0\o^7:)p=!#;7$$"0F!y`,>58!#9$"/sQ<NmS:!#:7$$"0Ee2pUgK"!#9$"0R")RK,8?"!#;7$$"/)*[<K^T8!#8$"0DqHH*4+))!#<7$$"0/y;V$Ge8!#9$"0aAD"[:Da!#<7$$"0(3dHVzs8!#9$"0eJF/nMf#!#<7$$"0OZLig%*Q"!#9$!0*eQIQK(f&!#=7$$"0Ctc"y709!#9$!0\RV[U%HM!#<7$$"012+oV1U"!#9$!0VUZ>oK='!#<7$$"0(oK()=<O9!#9$!0\_&RGS`))!#<7$$"0Dpen4?X"!#9$!077I'e-\6!#;7$$"/,Smuzo9!#8$!0YASn5#>9!#;7$$"0w*RE->%["!#9$!/dJ(Q7(e;!#:7$$"0"R/d[Q*\"!#9$!/h*3g&e()=!#:7$$"0jIrmFc^"!#9$!0OY5sXT7#!#;7$$"0`R!=^$=`"!#9$!03I)Q0/_B!#;7$$"0P,xE#QY:!#9$!0"GABi")\D!#;7$$"00;bT9Oc"!#9$!0oqTV+gx#!#;7$$"0Z4xZL!y:!#9$!0rS_Ho'eH!#;7$$"0/J0>)*[f"!#9$!0P2iz!)[;$!#;7$$"0MU^#R*4h"!#9$!0/SK_(QaL!#;7$$"0`E*)\l`i"!#9$!0^#z.%4x^$!#;7$$"0$GsOy_T;!#9$!0.z`A/\p$!#;7$$"0T&)*3tzd;!#9$!0%ppG3^mQ!#;7$$"0H'Gb%*)Rn"!#9$!0EI$=0tIS!#;7$$"0@Ywk,!*o"!#9$!0DWo'yExT!#;7$$"0z%H.fa/<!#9$!0\$H->MBV!#;7$$"0%Gs%z@1s"!#9$!0&fhaB]oW!#;7$$"/KcIhkO<!#8$!0$*o4qjtg%!#;7$$"0D;'H"HJv"!#9$!/L&)y`FWZ!#:7$$"02aTOZww"!#9$!0>KLH7+'[!#;7$$"0:4wX"*Ry"!#9$!0$Rj#y(*\)\!#;7$$"0E#[hES+=!#9$!0pw#\3)\5&!#;7$$"09Ot*y@;=!#9$!0ivhu5b@&!#;7$$"08=mkz0$=!#9$!0W4Uxb;J&!#;7$$"0b7c:dw%=!#9$!0%H>%fw3U&!#;7$$"0O"zXS7i=!#9$!0ftr=o"4b!#;7$$"0jp#yC&*y=!#9$!0WS:,9rg&!#;7$$"0/"[#\[Q*=!#9$!0yVo@\'*o&!#;7$$"/A'4y">5>!#8$!0WMHFAex&!#;7$$"0LZkXad#>!#9$!0GE?$ps`e!#;7$$"0>%ovC*>%>!#9$!0V:e^23$f!#;7$$"0j/j//p&>!#9$!0x)4Nk!z*f!#;7$$".-(*=))H(>!#7$!0H\;iHk1'!#;7$$"0rDV4&p*)>!#9$!0$Rp%4&[Lh!#;7$$"0i8R]QU+#!#9$!07KPzG&)='!#;7$$"0G)p%yX*>?!#9$!/6g9'*eWi!#:7$$"0O2F%H<O?!#9$!0[`Y33*)H'!#;7$$"/C+^![?0#!#8$!03/q)eg[j!#;7$$"/2cM!3u1#!#8$!0=+7QQNR'!#;7$$"0p'\xEY%3#!#9$!0jlDL'*)Rk!#;7$$"0d5F.(y*4#!#9$!0<nWUo%yk!#;7$$"0^>G0\h6#!#9$!0/]OyAl^'!#;7$$"0c7\Zv48#!#9$!0e6)[/H[l!#;7$$"0`J/n%=Z@!#9$!08:A\\,e'!#;7$$"0t--EOC;#!#9$!0z#o7GY2m!#;7$$"0[fgcy$y@!#9$!0_*G"oILj'!#;7$$"0lAS3lR>#!#9$!.QT4Xgl'!#97$$"/o%Qs#G5A!#8$!0j7g!\<xm!#;7$$"0%\R]&)*fA#!#9$!0h.%Rr.&p'!#;7$$"0aFHDq?C#!#9$!0F?zL`3r'!#;7$$"0v[O()3!eA!#9$!04K\cjTs'!#;7$$"0Uq2"[lsA!#9$!0oRpwpVt'!#;7$$"0-0vqS%*G#!#9$!0ohPc`Pu'!#;7$$"0d]O@aWI#!#9$!0))Hi`9,v'!#;7$$"0s,Euh/K#!#9$!0iv\gZ[v'!#;7$$"0lA.*HyNB!#9$!0ZSU9cuv'!#;7$$"0R3pE&z_B!#9$!0/wgY>#en!#;7$$"0#RS[m_nB!#9$!0w0+&G7dn!#;7$$"/f;)*oC%Q#!#8$!/6nK['Rv'!#:7$$"07t1f&[*R#!#9$!/\H(zp$\n!#:7$$"0u>aKahT#!#9$!0svU&)HDu'!#;7$$"0_1XiO0V#!#9$!0sZN1`^t'!#;7$$"0Y_VQ;pW#!#9$!0t2D]T^s'!#;7$$"0yFPUQFY#!#9$!0d-f/&*Qr'!#;7$$"0bZp7]&yC!#9$!0.:@#3;,n!#;7$$"/RwhOI%\#!#8$!0;>ivOqo'!#;7$$"0i(*o(yV4D!#9$!0PhANi@n'!#;7$$"0lcUO)zDD!#9$!0(yd7Epam!#;7$$"0d&*\4"RTD!#9$!0N5JkWnj'!#;7$$"0`f)4<!yb#!#9$!/36Yd`;m!#:7$$"0=$4oslsD!#9$!0o<Zn:rf'!#;7$$"0VM4Wv!*e#!#9$!0w-Nw_Wd'!#;7$$"0"ya*=,[g#!#9$!0.wei1;b'!#;7$$"0#*[7m)[?E!#9$!0#pJ-ouFl!#;7$$"/JC'Guoj#!#8$!0&)e4aF<]'!#;7$$"0DT]/m>l#!#9$!0TM>%z!oZ'!#;7$$"07-/2Aum#!#9$!0rXsst.X'!#;7$$"0(o"*et[%o#!#9$!0%e)Qnb,U'!#;7$$"0;^)z&Q**p#!#9$!0U@M2**=R'!#;7$$"0(*GuMSdr#!#9$!0-"e(*f:ij!#;7$$"0QI!Q."=t#!#9$!0M>!eo1Jj!#;7$$"0^9ly"eYF!#9$!01(4-@x,j!#;7$$"0<jAnFBw#!#9$!0-10.>)pi!#;7$$"0(fI*p`zx#!#9$!0[,Ad*RPi!#;7$$"0z$3"poYz#!#9$!00\N(G(>?'!#;7$$"/kBZ*=%4G!#8$!0WP_^'4qh!#;7$$"0=@3oPk#G!#9$!0s[$eejKh!#;7$$"0"HzvdzTG!#9$!01J2;K#)4'!#;7$$"0%y*z!=*p&G!#9$!00GnIfO1'!#;7$$"/()QCCLtG!#8$!0kr/&*>f-'!#;7$$"0@T>9R(*)G!#9$!0B3<)3Z()f!#;7$$"/nhDom/H!#8$!0AJEqF?&f!#;7$$"0p%fi$>0#H!#9$!0TcMTLR"f!#;7$$"0BE$*))*)f$H!#9$!/%G3jGj(e!#:7$$"0[9N5gF&H!#9$!0ib@A8^$e!#;7$$"/tS,5FnH!#8$!0QFij#4*z&!#;7$$"0z$=&HPR)H!#9$!0k'R9jLdd!#;7$$"0o4v[/'**H!#9$!0`kO"*Hxr&!#;7$$"0\V=N?^,$!#9$!/'zq(**=yc!#:7$$"/L;f&[1.$!#8$!0\LC#QKQc!#;7$$"0o0xM'[YI!#9$!09o!4%ztf&!#;7$$"0`O#QTFjI!#9$!0PGU%))o`b!#;7$$"0>O#)*omyI!#9$!0Nix/'Q8b!#;7$$"0M!))G:'Q4$!#9$!/%3J#=Rta!#:7$$"01n*QV55J!#9$!0`O#[jUIa!#;7$$"0'f()*y6j7$!#9$!0r4'3ZN(Q&!#;7$$"-X]EfTJ!#6$!0.$\/(zlM&!#;-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#"""-%'CURVESG6%7dw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"0P4yOPNs*!#:7$$"0p4!4F)))3$!#;$"0FV$3hF"[*!#:7$$"0FS')[;^q%!#;$"0iYsTbs?*!#:7$$"0*y*[rj?L'!#;$"0$4^n+3I*)!#:7$$"06')\My7&z!#;$"0-PhWMKl)!#:7$$"08y4e(\_%*!#;$"0w."f](fR)!#:7$$"0h$e9Bp+6!#:$"0b&Rn[GH")!#:7$$"/T')G7Xh7!#9$"0CE-!>V`y!#:7$$"0onse%p@9!#:$"0Ty7_:(yv!#:7$$"0B)zxX_'e"!#:$"0V_N`+nH(!#:7$$"0OwJ#pqJ<!#:$"0.+$pL,\q!#:7$$"0<Fx&y9&*=!#:$"/eK^GBrn!#97$$"0Jek*)f#f?!#:$"0j%zBLq$\'!#:7$$"02(*\D7u@#!#:$"0l#G#*3#yA'!#:7$$"0)p"yuH5O#!#:$"0f3By.z)f!#:7$$"0;hx$[!=`#!#:$"0=0:E]Zq&!#:7$$"0J\&RPZwE!#:$"0AQ^y!)oY&!#:7$$"0$>Lk!eZ%G!#:$"0tn/&pq#>&!#:7$$"0.Yk?=P*H!#:$"0u>_?hC&\!#:7$$"0od74^r:$!#:$"0MCY!oq"p%!#:7$$"0%*3h%yx7L!#:$"0eEu"fLYW!#:7$$"0cx%Q!e^Z$!#:$"/'>iqRN>%!#97$$".#oWPFCO!#8$"0o9T'*QW'R!#:7$$"0nb'y^6&y$!#:$"0&48V<u?P!#:7$$"0"G*[A%=_R!#:$"0D!e:O_rM!#:7$$"0%=x?$=w4%!#:$"0">PX)[zD$!#:7$$"0X='G6paU!#:$"03\8ZD4.$!#:7$$"/$4(3F'pT%!#9$"0ovwW#[+G!#:7$$"0kf;z8dd%!#:$"0&p`m*z"zD!#:7$$"/FCFOJHZ!#9$"0)Ru1l0pB!#:7$$"0i-m0g)**[!#:$"0WlH7l/9#!#:7$$"0RT(3O5`]!#:$"0b3*4fQR>!#:7$$"/3$)4Qs;_!#9$"0ckMc"HH<!#:7$$"/8wI!))\O&!#9$"0uKJE&3V:!#:7$$"0(4&f)*zq_&!#:$"0t$3bp6W8!#:7$$"0*Hm$)efzc!#:$"0^b_M^8;"!#:7$$"0VIA%*=!Re!#:$"0KwVX>(\(*!#;7$$"0@i=7%)[*f!#:$"0J_v@eS(z!#;7$$"0j.,_g!eh!#:$".'QkM$[;'!#97$$"04&e&y=_J'!#:$"0O8G]q2Z%!#;7$$"0664"e$fZ'!#:$"0f=v."z(y#!#;7$$"0;B"=?KNm!#:$"0zUNca$o6!#;7$$"0"*R$*Q"y"y'!#:$!0Oz.=-/w#!#<7$$"0(eoc.k\p!#:$!/"y&**H*)z=!#:7$$"0PT"=ax*4(!#:$!0y8%epmnK!#;7$$"0"Hl22&)fs!#:$!0J#[%p9()p%!#;7$$"0<i[=jIT(!#:$!0d-Sao9-'!#;7$$"0c>2&f=$e(!#:$!/KCp>`Ou!#:7$$"0)[nl(*\Ix!#:$!0@-Dzojh)!#;7$$"0p%HjAq(*y!#:$!0*R;jz_/**!#;7$$"0$pO)=*3]!)!#:$!0COsHdJ5"!#:7$$"06Le`wn@)!#:$!0OQa4R8A"!#:7$$"0#4qE&*fg$)!#:$!0MnRKc!>8!#:7$$"0Cg^7(RC&)!#:$!0\@X"=eD9!#:7$$"0^8*=vh#o)!#:$!0H0g*QqB:!#:7$$"0>66bM2%))!#:$!0A"*>$p6<;!#:7$$"0ru#**)p#)**)!#:$!0"o`u^h0<!#:7$$"0$=h]?h\"*!#:$!0uc*)Q$Q'y"!#:7$$"0A-U#p@8$*!#:$!0N_cAl!p=!#:7$$"0O"fJU9p%*!#:$!0O)z*=XM%>!#:7$$"0(4B!Q]Kj*!#:$!0*4p8"Hr,#!#:7$$"0[nD'f!=y*!#:$!0iKT"=#)z?!#:7$$"0-!)4p()f%**!#:$!0;pC(RuW@!#:7$$"0Q6x^C.,"!#9$!0.")p.)p-A!#:7$$"0\7%*)>,E5!#9$!0;6E8ckD#!#:7$$"0n1Wh(RU5!#9$!0rBk3e$3B!#:7$$"0"[?t$*[d5!#9$!0[ivp!R_B!#:7$$"0pl&)RXH2"!#9$!0i"4g&3QR#!#:7$$"0W!3(o5+4"!#9$!0Y!\->INC!#:7$$"0t9!3>Y06!#9$!0."3o]6pC!#:7$$"0`#fvOE@6!#9$!0'3dnV3+D!#:7$$"0%R>mOLP6!#9$!0'zC<w"z_#!#:7$$"02yY60@:"!#9$!0tU!3)=.b#!#:7$$"0tE/+^y;"!#9$!0L9)o%>4d#!#:7$$"0apu-xM="!#9$!0)HDC'*4)e#!#:7$$"0OZ#>?>+7!#9$!0F+&o<(Hg#!#:7$$"0'**RvA%\@"!#9$!0$3:Rd;8E!#:7$$"0v%)*35'>B"!#9$!0rrp75;i#!#:7$$"0[cR5>tC"!#9$!0W]t!fEEE!#:7$$"0Thh8:DE"!#9$!0Oo%[g=GE!#:7$$"0E_Dvb)y7!#9$!0drEueti#!#:7$$"0y/,Zi_H"!#9$!0.b0k3Oi#!#:7$$"0F!y`,>58!#9$!0&)f`'zt<E!#:7$$"0Ee2pUgK"!#9$!0i+#)\C!4E!#:7$$"/)*[<K^T8!#8$!04%eaB9)f#!#:7$$"0/y;V$Ge8!#9$!0n"*)p%)y$e#!#:7$$"0(3dHVzs8!#9$!0mq.y*HpD!#:7$$"0OZLig%*Q"!#9$!0auC%GQ]D!#:7$$"0Ctc"y709!#9$!0XR](yYID!#:7$$"012+oV1U"!#9$!0)f7DLz3D!#:7$$"0(oK()=<O9!#9$!0OoeFQ_[#!#:7$$"0Dpen4?X"!#9$!0a^;>x$fC!#:7$$"/,Smuzo9!#8$!0mIo3M+V#!#:7$$"0w*RE->%["!#9$!0md+3n9S#!#:7$$"0"R/d[Q*\"!#9$!0c&3n]yrB!#:7$$"0jIrmFc^"!#9$!0/P604&QB!#:7$$"0`R!=^$=`"!#9$!0.QkB)z.B!#:7$$"0P,xE#QY:!#9$!0='**f>VrA!#:7$$"00;bT9Oc"!#9$!0F2V_'pJA!#:7$$"0Z4xZL!y:!#9$!0\'f/TN(>#!#:7$$"0/J0>)*[f"!#9$!0Giz90g:#!#:7$$"0MU^#R*4h"!#9$!084lALa6#!#:7$$"0`E*)\l`i"!#9$!0.CrLd$y?!#:7$$"0$GsOy_T;!#9$!0"oG"Gud.#!#:7$$"0T&)*3tzd;!#9$!0H1$oY.#*>!#:7$$"0H'Gb%*)Rn"!#9$!0yq$*f2x%>!#:7$$"0@Ywk,!*o"!#9$!/45oj'f!>!#97$$"0z%H.fa/<!#9$!0$4TFa:i=!#:7$$"0%Gs%z@1s"!#9$!0Mt1>!G;=!#:7$$"/KcIhkO<!#8$!//!*4g/q<!#97$$"0D;'H"HJv"!#9$!0PC4QI?s"!#:7$$"02aTOZww"!#9$!/A)=?3%z;!#97$$"0:4wX"*Ry"!#9$!/9B)>;6j"!#97$$"0E#[hES+=!#9$!0D9y`eBe"!#:7$$"09Ot*y@;=!#9$!0y5&es<N:!#:7$$"08=mkz0$=!#9$!0Ps8(R@#\"!#:7$$"0b7c:dw%=!#9$!0n3jWP5W"!#:7$$"0O"zXS7i=!#9$!083abawR"!#:7$$"0jp#yC&*y=!#9$!0Ck,'*4sM"!#:7$$"0/"[#\[Q*=!#9$!/fSH4i-8!#97$$"/A'4y">5>!#8$!03^ds;QD"!#:7$$"0LZkXad#>!#9$!04bh@-v?"!#:7$$"0>%ovC*>%>!#9$!0Ht\8*Qf6!#:7$$"0j/j//p&>!#9$!0HNCFNa6"!#:7$$".-(*=))H(>!#7$!0'*zya6$o5!#:7$$"0rDV4&p*)>!#9$!0kl"R9s>5!#:7$$"0i8R]QU+#!#9$!0&3X'3^vx*!#;7$$"0G)p%yX*>?!#9$!0v>qb5"G$*!#;7$$"0O2F%H<O?!#9$!0J:Ux*Ho))!#;7$$"/C+^![?0#!#8$!0siL$oBB%)!#;7$$"/2cM!3u1#!#8$!/4bWFT(*z!#:7$$"0p'\xEY%3#!#9$!0%)f>.v/`(!#;7$$"0d5F.(y*4#!#9$!0%y(RWjk6(!#;7$$"0^>G0\h6#!#9$!0H,jA40o'!#;7$$"0c7\Zv48#!#9$!/vh5i8"H'!#:7$$"0`J/n%=Z@!#9$!0,@3Dd=(e!#;7$$"0t--EOC;#!#9$!0tCOGvO[&!#;7$$"0[fgcy$y@!#9$!0DigG%o%3&!#;7$$"0lAS3lR>#!#9$!0YR!GP[,Z!#;7$$"/o%Qs#G5A!#8$!0Y=EJ+yI%!#;7$$"0%\R]&)*fA#!#9$!0#43kv0OR!#;7$$"0aFHDq?C#!#9$!0yCjy%fjN!#;7$$"0v[O()3!eA!#9$!0uP[#e0-K!#;7$$"0Uq2"[lsA!#9$!/U7%\To(G!#:7$$"0-0vqS%*G#!#9$!0pi#oo[7D!#;7$$"0d]O@aWI#!#9$!0HMtX)G%>#!#;7$$"0s,Euh/K#!#9$!0Z&H9^6j=!#;7$$"0lA.*HyNB!#9$!0s"fO[/a:!#;7$$"0R3pE&z_B!#9$!0HZ)zm+?7!#;7$$"0#RS[m_nB!#9$!0TV,H"*eQ*!#<7$$"/f;)*oC%Q#!#8$!0j?*pLZ!G'!#<7$$"07t1f&[*R#!#9$!/m`lbyKN!#;7$$"0u>aKahT#!#9$!0-Erd;y<'!#=7$$"0_1XiO0V#!#9$"0)[#)\d9@=!#<7$$"0Y_VQ;pW#!#9$"0rOI(zk7X!#<7$$"0yFPUQFY#!#9$"0_JrnV`-(!#<7$$"0bZp7]&yC!#9$"0b5OGp3X*!#<7$$"/RwhOI%\#!#8$"0QyH-d#y6!#;7$$"0i(*o(yV4D!#9$"0i%[c()H%R"!#;7$$"0lcUO)zDD!#9$"0-xd*)*4>;!#;7$$"0d&*\4"RTD!#9$"0#)f*H?$\#=!#;7$$"0`f)4<!yb#!#9$"0.k2uLF.#!#;7$$"0=$4oslsD!#9$"0'=nAg38A!#;7$$"0VM4Wv!*e#!#9$"0nS21%*QS#!#;7$$"0"ya*=,[g#!#9$"001:cQ$yD!#;7$$"0#*[7m)[?E!#9$"0R!=8LKWF!#;7$$"/JC'Guoj#!#8$"00*RK)4#4H!#;7$$"0DT]/m>l#!#9$"/A:0n[`I!#:7$$"07-/2Aum#!#9$"0T?:Y(y$>$!#;7$$"0(o"*et[%o#!#9$"0')p#zi.SL!#;7$$"0;^)z&Q**p#!#9$"/v%H%frkM!#:7$$"0(*GuMSdr#!#9$"0BEbWEZe$!#;7$$"0QI!Q."=t#!#9$"0*)z&396*p$!#;7$$"0^9ly"eYF!#9$"0$4#Q'4a(z$!#;7$$"0<jAnFBw#!#9$"0;y,?'\&*Q!#;7$$"0(fI*p`zx#!#9$"0:dj8:d)R!#;7$$"0z$3"poYz#!#9$"0W'[&Q^Y2%!#;7$$"/kBZ*=%4G!#8$"0w#=KVvYT!#;7$$"0=@3oPk#G!#9$"0=9AzhEA%!#;7$$"0"HzvdzTG!#9$"/Qmivf%G%!#:7$$"0%y*z!=*p&G!#9$"0iB^2z)RV!#;7$$"/()QCCLtG!#8$"0TB*e"3GR%!#;7$$"0@T>9R(*)G!#9$"/cPmKJRW!#:7$$"/nhDom/H!#8$"0f;bj$)fZ%!#;7$$"0p%fi$>0#H!#9$"0=Kb!G=4X!#;7$$"0BE$*))*)f$H!#9$"0')4))*)=g`%!#;7$$"0[9N5gF&H!#9$"0lt)ep1fX!#;7$$"/tS,5FnH!#8$"0C5eQsSd%!#;7$$"0z$=&HPR)H!#9$"01/)=W"ee%!#;7$$"0o4v[/'**H!#9$"0')[#G6l"f%!#;7$$"0\V=N?^,$!#9$"0GGlY8Ef%!#;7$$"/L;f&[1.$!#8$"0&e-PC#*)e%!#;7$$"0o0xM'[YI!#9$"00=#GJ^!e%!#;7$$"0`O#QTFjI!#9$"0W33+]mc%!#;7$$"0>O#)*omyI!#9$"0xU,L<'\X!#;7$$"0M!))G:'Q4$!#9$"08%*z7#*)GX!#;7$$"01n*QV55J!#9$"/OU_Uf-X!#:7$$"0'f()*y6j7$!#9$"0kuf<[AZ%!#;7$$"-X]EfTJ!#6$"/a9U&G+W%!#:-%&COLORG6&%$RGBG$""!!""$""!!""$"#5!""-%*LINESTYLEG6#"""-%'CURVESG6%7dw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"0naZ1Wor*!#:7$$"0p4!4F)))3$!#;$"03:^&4He%*!#:7$$"0FS')[;^q%!#;$"0f:>(>.b"*!#:7$$"0*y*[rj?L'!#;$"0'yOvO_P))!#:7$$"06')\My7&z!#;$"/^&Ryx/^)!#97$$"08y4e(\_%*!#;$"0T9')\2%)>)!#:7$$"0h$e9Bp+6!#:$"0&fN3DFny!#:7$$"/T')G7Xh7!#9$"/lI^[F<v!#97$$"0onse%p@9!#:$"0PHi]?=;(!#:7$$"0B)zxX_'e"!#:$"/$o*4JV!z'!#97$$"0OwJ#pqJ<!#:$"/t/B0Nfk!#97$$"0<Fx&y9&*=!#:$"0>)HG::$3'!#:7$$"0Jek*)f#f?!#:$"0QkB[.Fq&!#:7$$"02(*\D7u@#!#:$"0/R(*\fWL&!#:7$$"0)p"yuH5O#!#:$"0p&4OQW**\!#:7$$"0;hx$[!=`#!#:$"0pOBrP7g%!#:7$$"0J\&RPZwE!#:$"0&f8r$yZE%!#:7$$"0$>Lk!eZ%G!#:$"0e%R'3r_(Q!#:7$$"0.Yk?=P*H!#:$"0rw?X*)G`$!#:7$$"0od74^r:$!#:$"01q'=yggJ!#:7$$"0%*3h%yx7L!#:$"/o>#Q$35G!#97$$"0cx%Q!e^Z$!#:$"0Z26T,#\C!#:7$$".#oWPFCO!#8$"06))*RJwA@!#:7$$"0nb'y^6&y$!#:$"0l7B^+mx"!#:7$$"0"G*[A%=_R!#:$"/O+dh@C9!#97$$"0%=x?$=w4%!#:$"09lL=ER7"!#:7$$"0X='G6paU!#:$"0r"eEl^o!)!#;7$$"/$4(3F'pT%!#9$"06KPcBo([!#;7$$"0kf;z8dd%!#:$"0u;D$3mT=!#;7$$"/FCFOJHZ!#9$!0ON$oAz35!#;7$$"0i-m0g)**[!#:$!0zXIS?+2%!#;7$$"0RT(3O5`]!#:$!0^R^:XSs'!#;7$$"/3$)4Qs;_!#9$!0bZ'3(3MX*!#;7$$"/8wI!))\O&!#9$!/j8e>4$="!#97$$"0(4&f)*zq_&!#:$!0LFt:\BV"!#:7$$"0*Hm$)efzc!#:$!0H-")z'\c;!#:7$$"0VIA%*=!Re!#:$!0.['RC%)z=!#:7$$"0@i=7%)[*f!#:$!0>8T]Bs3#!#:7$$"0j.,_g!eh!#:$!0XR<tiDH#!#:7$$"04&e&y=_J'!#:$!0*oUX`%)yC!#:7$$"0664"e$fZ'!#:$!0i.8h=wl#!#:7$$"0;B"=?KNm!#:$!0$[J<a:BG!#:7$$"0"*R$*Q"y"y'!#:$!0wj?:J\'H!#:7$$"0(eoc.k\p!#:$!0%)=J6g_6$!#:7$$"0PT"=ax*4(!#:$!0;#Hz_uQK!#:7$$"0"Hl22&)fs!#:$!03]@pc!fL!#:7$$"0<i[=jIT(!#:$!0)3]C%>LY$!#:7$$"0c>2&f=$e(!#:$!0%3cI5tmN!#:7$$"0)[nl(*\Ix!#:$!0N)GJE*ek$!#:7$$"0p%HjAq(*y!#:$!0j&*3FEUs$!#:7$$"0$pO)=*3]!)!#:$!02uNz>^y$!#:7$$"06Le`wn@)!#:$!0f6N&3ZSQ!#:7$$"0#4qE&*fg$)!#:$!0ko:8g*yQ!#:7$$"0Cg^7(RC&)!#:$!0m**='[c7R!#:7$$"0^8*=vh#o)!#:$!0#3")4S#\$R!#:7$$"0>66bM2%))!#:$!0<v8f7w%R!#:7$$"0ru#**)p#)**)!#:$!0Dr)*pC4&R!#:7$$"0$=h]?h\"*!#:$!0Crmm(fXR!#:7$$"0A-U#p@8$*!#:$!0>N9(QwIR!#:7$$"0O"fJU9p%*!#:$!0'f'Gga"3R!#:7$$"0(4B!Q]Kj*!#:$!0)ou&pod(Q!#:7$$"0[nD'f!=y*!#:$!/PeAG:RQ!#97$$"0-!)4p()f%**!#:$!0&[CO4'4z$!#:7$$"0Q6x^C.,"!#9$!0R9\_Pvt$!#:7$$"0\7%*)>,E5!#9$!0yV)yh]xO!#:7$$"0n1Wh(RU5!#9$!/lJ#))))zg$!#97$$"0"[?t$*[d5!#9$!0s"Q%3O"QN!#:7$$"0pl&)RXH2"!#9$!0&*)>%)=9hM!#:7$$"0W!3(o5+4"!#9$!0*Rp<!>,P$!#:7$$"0t9!3>Y06!#9$!0^*3b*GEG$!#:7$$"0`#fvOE@6!#9$!0"Q5Ds_)=$!#:7$$"0%R>mOLP6!#9$!0b+T(yR)3$!#:7$$"02yY60@:"!#9$!0B0QQSF*H!#:7$$"0tE/+^y;"!#9$!0oH'G?H()G!#:7$$"0apu-xM="!#9$!0:]"fJVzF!#:7$$"0OZ#>?>+7!#9$!/7Ts%)*3m#!#97$$"0'**RvA%\@"!#9$!0y7;R.Rb#!#:7$$"0v%)*35'>B"!#9$!0>&*=KU!GC!#:7$$"0[cR5>tC"!#9$!0cT=pwDJ#!#:7$$"0Thh8:DE"!#9$!02v)\c(o>#!#:7$$"0E_Dvb)y7!#9$!0Af;=v62#!#:7$$"0y/,Zi_H"!#9$!/'ohfpR%>!#97$$"0F!y`,>58!#9$!0\DC@`w#=!#:7$$"0Ee2pUgK"!#9$!0o(*=D=Qq"!#:7$$"/)*[<K^T8!#8$!0O%=PE$He"!#:7$$"0/y;V$Ge8!#9$!0>b[(>;_9!#:7$$"0(3dHVzs8!#9$!0Tk:W![R8!#:7$$"0OZLig%*Q"!#9$!0K7*Qy(3@"!#:7$$"0Ctc"y709!#9$!0hq'oP."4"!#:7$$"012+oV1U"!#9$!/l$yFmet*!#:7$$"0(oK()=<O9!#9$!0t3]\<]d)!#;7$$"0Dpen4?X"!#9$!0))z-*p?3u!#;7$$"/,Smuzo9!#8$!0;@x;*p#>'!#;7$$"0w*RE->%["!#9$!0@M1Vg%*4&!#;7$$"0"R/d[Q*\"!#9$!0B]f1)*>/%!#;7$$"0jIrmFc^"!#9$!0DMt*oAPH!#;7$$"0`R!=^$=`"!#9$!0&=%)=r-j=!#;7$$"0P,xE#QY:!#9$!0z(*=3p?C*!#<7$$"00;bT9Oc"!#9$"0pI%Re$Gb"!#<7$$"0Z4xZL!y:!#9$"0XV-v$>I5!#;7$$"0/J0>)*[f"!#9$"0CTy4/%>?!#;7$$"0MU^#R*4h"!#9$"02aO!e0GH!#;7$$"0`E*)\l`i"!#9$"0)=5ACC4P!#;7$$"0$GsOy_T;!#9$"0_P0n\Hb%!#;7$$"0T&)*3tzd;!#9$"083-36UO&!#;7$$"0H'Gb%*)Rn"!#9$"0za1t**H8'!#;7$$"0@Ywk,!*o"!#9$"0dFvC<3"o!#;7$$"0z%H.fa/<!#9$"0r!Hyuywu!#;7$$"0%Gs%z@1s"!#9$"0yz!46yE")!#;7$$"/KcIhkO<!#8$"0y'>&Q(>N()!#;7$$"0D;'H"HJv"!#9$"0zVmI-(>$*!#;7$$"02aTOZww"!#9$"0"3XP9v*z*!#;7$$"0:4wX"*Ry"!#9$"0v)Qg.7I5!#:7$$"0E#[hES+=!#9$"0l!>E$Hj2"!#:7$$"09Ot*y@;=!#9$"06$ynA&p6"!#:7$$"08=mkz0$=!#9$"0pU^MT0:"!#:7$$"0b7c:dw%=!#9$"0"yw@-V'="!#:7$$"0O"zXS7i=!#9$"0YbZ!fU87!#:7$$"0jp#yC&*y=!#9$"0#)>[<\4C"!#:7$$"0/"[#\[Q*=!#9$"0#3`oD)=E"!#:7$$"/A'4y">5>!#8$"0vvGE47G"!#:7$$"0LZkXad#>!#9$"0r%pE$RhH"!#:7$$"0>%ovC*>%>!#9$"0v=!*)G=38!#:7$$"0j/j//p&>!#9$"0/)[0k8;8!#:7$$".-(*=))H(>!#7$"/>pvcY@8!#97$$"0rDV4&p*)>!#9$"0#fC<"HNK"!#:7$$"0i8R]QU+#!#9$"0*zUV,`A8!#:7$$"0G)p%yX*>?!#9$"0z'[(>@'=8!#:7$$"0O2F%H<O?!#9$"0avY')*f68!#:7$$"/C+^![?0#!#8$"0:_:F0>I"!#:7$$"/2cM!3u1#!#8$"0dO!)3p**G"!#:7$$"0p'\xEY%3#!#9$"00Wv]&*QF"!#:7$$"0d5F.(y*4#!#9$"0;v(4V.d7!#:7$$"0^>G0\h6#!#9$"0=EKSFmB"!#:7$$"0c7\Zv48#!#9$"0wq8u(4;7!#:7$$"0`J/n%=Z@!#9$"06c*H`b">"!#:7$$"0t--EOC;#!#9$"02nBbtl;"!#:7$$"0[fgcy$y@!#9$"0Wl<!=iQ6!#:7$$"0lAS3lR>#!#9$"/#zf(Rf46!#97$$"/o%Qs#G5A!#8$"/v(pmIv2"!#97$$"0%\R]&)*fA#!#9$"0cv&*G`^/"!#:7$$"0aFHDq?C#!#9$"0;feu[1,"!#:7$$"0v[O()3!eA!#9$"0[Tr)pg^(*!#;7$$"0Uq2"[lsA!#9$"0dCAn7aT*!#;7$$"0-0vqS%*G#!#9$"09)H!zW%>!*!#;7$$"0d]O@aWI#!#9$"0%zNEjtc')!#;7$$"0s,Euh/K#!#9$"0K=<^pAE)!#;7$$"0lA.*HyNB!#9$"0l^n&))Gyy!#;7$$"0R3pE&z_B!#9$"/tiK'3eW(!#:7$$"0#RS[m_nB!#9$"0_BCS?r1(!#;7$$"/f;)*oC%Q#!#8$"0dm>FNPj'!#;7$$"07t1f&[*R#!#9$"0uE*35WOi!#;7$$"0u>aKahT#!#9$"0-*>'HV/!e!#;7$$"0_1XiO0V#!#9$"0KH'*o:RU&!#;7$$"0Y_VQ;pW#!#9$"0QE\lxc*\!#;7$$"0yFPUQFY#!#9$"0o&*)\R`$e%!#;7$$"0bZp7]&yC!#9$"0mI)yt/uT!#;7$$"/RwhOI%\#!#8$"/[OhMHpP!#:7$$"0i(*o(yV4D!#9$"0LXm3VUQ$!#;7$$"0lcUO)zDD!#9$"0,ER&G)H(H!#;7$$"0d&*\4"RTD!#9$"0.&)>N#f'e#!#;7$$"0`f)4<!yb#!#9$"0w**3yul=#!#;7$$"0=$4oslsD!#9$"0z')G(*f4$=!#;7$$"0VM4Wv!*e#!#9$"04@"HiyX9!#;7$$"0"ya*=,[g#!#9$".$\m&*=&3"!#97$$"0#*[7m)[?E!#9$"/nI@YfTt!#;7$$"/JC'Guoj#!#8$"0YfO?`Lx$!#<7$$"0DT]/m>l#!#9$"0%eEjkp.e!#=7$$"07-/2Aum#!#9$!0)or3$*\#f#!#<7$$"0(o"*et[%o#!#9$!0_)pGv$o(f!#<7$$"0;^)z&Q**p#!#9$!0&z*4:)eH*)!#<7$$"0(*GuMSdr#!#9$!0'eKc*GO="!#;7$$"0QI!Q."=t#!#9$!/XGzt>n9!#:7$$"0^9ly"eYF!#9$!0"*fam/pr"!#;7$$"0<jAnFBw#!#9$!0<Q3E;8(>!#;7$$"0(fI*p`zx#!#9$!//kMZf6A!#:7$$"0z$3"poYz#!#9$!0t#>**o)\X#!#;7$$"/kBZ*=%4G!#8$!0'[g9^%zl#!#;7$$"0=@3oPk#G!#9$!/6I&zK#yG!#:7$$"0"HzvdzTG!#9$!0^`2R">kI!#;7$$"0%y*z!=*p&G!#9$!0d-DVrhB$!#;7$$"/()QCCLtG!#8$!0HxQAkxS$!#;7$$"0@T>9R(*)G!#9$!0mNVophc$!#;7$$"/nhDom/H!#8$!09v6$3E)p$!#;7$$"0p%fi$>0#H!#9$!0d_W:hg#Q!#;7$$"0BE$*))*)f$H!#9$!01-w;&\QR!#;7$$"0[9N5gF&H!#9$!0.>$)37o/%!#;7$$"/tS,5FnH!#8$!0_]g5"HHT!#;7$$"0z$=&HPR)H!#9$!0"yRm'Q8@%!#;7$$"0o4v[/'**H!#9$!0H-d`1jF%!#;7$$"0\V=N?^,$!#9$!0(*\MG]#HV!#;7$$"/L;f&[1.$!#8$!0R(>)y76P%!#;7$$"0o0xM'[YI!#9$!0*)R\m/ES%!#;7$$"0`O#QTFjI!#9$!0)44YC%RU%!#;7$$"0>O#)*omyI!#9$!0)R'e_7HV%!#;7$$"0M!))G:'Q4$!#9$!04yCy:@V%!#;7$$"01n*QV55J!#9$!0)*\mJ))4U%!#;7$$"0'f()*y6j7$!#9$!0r-Ahe'*R%!#;7$$"-X]EfTJ!#6$!0l$Hw4`qV!#;-%&COLORG6&%$RGBG$"#5!""$""!!""$""!!""-%*LINESTYLEG6#"""-%'CURVESG6%7dw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"0Uy7%y[2(*!#:7$$"0p4!4F)))3$!#;$"0([:$=$=E%*!#:7$$"0FS')[;^q%!#;$"0>Qm/u@3*!#:7$$"0*y*[rj?L'!#;$"0nO%R4j3()!#:7$$"06')\My7&z!#;$"0$HKuP97$)!#:7$$"08y4e(\_%*!#;$"02$o(pbY#z!#:7$$"0h$e9Bp+6!#:$"0Kv%3(4a](!#:7$$"/T')G7Xh7!#9$"0(*e9-J[0(!#:7$$"0onse%p@9!#:$"/9i'op3f'!#97$$"0B)zxX_'e"!#:$"0ZS0x$o+h!#:7$$"0OwJ#pqJ<!#:$"04)Gww1gc!#:7$$"0<Fx&y9&*=!#:$"0<X3G\j:&!#:7$$"0Jek*)f#f?!#:$"0dbXl5Zk%!#:7$$"02(*\D7u@#!#:$"0kP&=qL[T!#:7$$"0)p"yuH5O#!#:$"0BtomRlp$!#:7$$"0;hx$[!=`#!#:$"/%yar%=gJ!#97$$"0J\&RPZwE!#:$"/^/U&)H3F!#97$$"0$>Lk!eZ%G!#:$"0T2[&e`(=#!#:7$$"0.Yk?=P*H!#:$"0w6i@FEt"!#:7$$"0od74^r:$!#:$"0c()*pI&=C"!#:7$$"0%*3h%yx7L!#:$"06%fEHNUy!#;7$$"0cx%Q!e^Z$!#:$"0)e\8w"\=$!#;7$$".#oWPFCO!#8$!0U'RF'=;t*!#<7$$"0nb'y^6&y$!#:$!04=,DXqJ&!#;7$$"0"G*[A%=_R!#:$!0kW7me+m*!#;7$$"0%=x?$=w4%!#:$!0fOPL%)*G8!#:7$$"0X='G6paU!#:$!0Z**G/VUq"!#:7$$"/$4(3F'pT%!#9$!0z/CigE2#!#:7$$"0kf;z8dd%!#:$!0.I*QGE8C!#:7$$"/FCFOJHZ!#9$!0cT:_HMs#!#:7$$"0i-m0g)**[!#:$!0(="p"[vWI!#:7$$"0RT(3O5`]!#:$!0AF9FRAJ$!#:7$$"/3$)4Qs;_!#9$!0KA6\v^d$!#:7$$"/8wI!))\O&!#9$!0Kk!R%=Hz$!#:7$$"0(4&f)*zq_&!#:$!0RQ1l4%3S!#:7$$"0*Hm$)efzc!#:$!/O#3s>&*=%!#97$$"0VIA%*=!Re!#:$!0`y*[WMcV!#:7$$"0@i=7%)[*f!#:$!0w??!eE(\%!#:7$$"0j.,_g!eh!#:$!0N=u?Q9i%!#:7$$"04&e&y=_J'!#:$!0s]$*)=m=Z!#:7$$"0664"e$fZ'!#:$!04L-/)p&z%!#:7$$"0;B"=?KNm!#:$!0haUPE,&[!#:7$$"0"*R$*Q"y"y'!#:$!/>G8.D")[!#97$$"0(eoc.k\p!#:$!0Y;`9B_*[!#:7$$"0PT"=ax*4(!#:$!0&*)G3fj))[!#:7$$"0"Hl22&)fs!#:$!0,X)z=Ti[!#:7$$"0<i[=jIT(!#:$!0\?$yWT>[!#:7$$"0c>2&f=$e(!#:$!0;S;Y6?v%!#:7$$"0)[nl(*\Ix!#:$!08zu:!oxY!#:7$$"0p%HjAq(*y!#:$!0(oxcEFwX!#:7$$"0$pO)=*3]!)!#:$!0c#*p=@*oW!#:7$$"06Le`wn@)!#:$!0w6'4o>OV!#:7$$"0#4qE&*fg$)!#:$!0U"Q"*4m4U!#:7$$"0Cg^7(RC&)!#:$!0'yoc](H0%!#:7$$"0^8*=vh#o)!#:$!0Wybc-**)Q!#:7$$"0>66bM2%))!#:$!00p&z6X;P!#:7$$"0ru#**)p#)**)!#:$!0-"\$)oAMN!#:7$$"0$=h]?h\"*!#:$!0V))=lj7N$!#:7$$"0A-U#p@8$*!#:$!0fH8c>e9$!#:7$$"0O"fJU9p%*!#:$!0,bm%yiVH!#:7$$"0(4B!Q]Kj*!#:$!0Htdt(=DF!#:7$$"0[nD'f!=y*!#:$!0aYH?#QBD!#:7$$"0-!)4p()f%**!#:$!0f&*p<$)oH#!#:7$$"0Q6x^C.,"!#9$!0$)*pqL]x?!#:7$$"0\7%*)>,E5!#9$!0c,9MOs&=!#:7$$"0n1Wh(RU5!#9$!0vre_Qmi"!#:7$$"0"[?t$*[d5!#9$!0a*Q)pFYT"!#:7$$"0pl&)RXH2"!#9$!0)y/^Sp)>"!#:7$$"0W!3(o5+4"!#9$!0eR.*H@E'*!#;7$$"0t9!3>Y06!#9$!0["[uI?=v!#;7$$"0`#fvOE@6!#9$!0c_fNk!*R&!#;7$$"0%R>mOLP6!#9$!0E"*z#Q\*G$!#;7$$"02yY60@:"!#9$!0kw9y&4(R"!#;7$$"0tE/+^y;"!#9$"0_r;N8tk&!#<7$$"0apu-xM="!#9$"0()o<)*)H\C!#;7$$"0OZ#>?>+7!#9$"0jdQ9!H!R%!#;7$$"0'**RvA%\@"!#9$"0KR0eKS.'!#;7$$"0v%)*35'>B"!#9$"0dqdQ`[%y!#;7$$"0[cR5>tC"!#9$"0O9K4TfR*!#;7$$"0Thh8:DE"!#9$"0iSA%)e\3"!#:7$$"0E_Dvb)y7!#9$"0'HMqb$>B"!#:7$$"0y/,Zi_H"!#9$"0ZM&e;Yp8!#:7$$"0F!y`,>58!#9$"0=+`A>c["!#:7$$"0Ee2pUgK"!#9$"0&>)\5F%*f"!#:7$$"/)*[<K^T8!#8$".CH,r3q"!#87$$"0/y;V$Ge8!#9$"0"[>9p)**z"!#:7$$"0(3dHVzs8!#9$"0r!*H9&ew=!#:7$$"0OZLig%*Q"!#9$"0Mpob\S&>!#:7$$"0Ctc"y709!#9$"0N>:9Wm,#!#:7$$"012+oV1U"!#9$"0xx[gE*o?!#:7$$"0(oK()=<O9!#9$"0T3x'Gn6@!#:7$$"0Dpen4?X"!#9$"0(47uU`X@!#:7$$"/,Smuzo9!#8$"/MP&*3)3<#!#97$$"0w*RE->%["!#9$"/%\_7)y%=#!#97$$"0"R/d[Q*\"!#9$"/j0_y(**=#!#97$$"0jIrmFc^"!#9$"0LtUW6k=#!#:7$$"0`R!=^$=`"!#9$"09#G0Uxt@!#:7$$"0P,xE#QY:!#9$"0^YG&**)\:#!#:7$$"00;bT9Oc"!#9$"0'G(zf2S7#!#:7$$"0Z4xZL!y:!#9$"0LfyWG64#!#:7$$"0/J0>)*[f"!#9$"0`*y@T/X?!#:7$$"0MU^#R*4h"!#9$"0Q#oXD"Q*>!#:7$$"0`E*)\l`i"!#9$"0LJ*HsWU>!#:7$$"0$GsOy_T;!#9$"0[q7*)e(y=!#:7$$"0T&)*3tzd;!#9$"0Q)Rv/x3=!#:7$$"0H'Gb%*)Rn"!#9$"0cj,4RPt"!#:7$$"0@Ywk,!*o"!#9$"0(\TB#3)f;!#:7$$"0z%H.fa/<!#9$"0<"f\jDz:!#:7$$"0%Gs%z@1s"!#9$"/)f/nE@\"!#97$$"/KcIhkO<!#8$"0x(*H3$)=S"!#:7$$"0D;'H"HJv"!#9$"0lVD3DgI"!#:7$$"02aTOZww"!#9$"0F,Kh`%>7!#:7$$"0:4wX"*Ry"!#9$"0td*p20?6!#:7$$"0E#[hES+=!#9$"0F.,cW'=5!#:7$$"09Ot*y@;=!#9$"/C9>@k)>*!#:7$$"08=mkz0$=!#9$"0>@NVuiH)!#;7$$"0b7c:dw%=!#9$"0RcL#=5@s!#;7$$"0O"zXS7i=!#9$"/1GU%=?J'!#:7$$"0jp#yC&*y=!#9$"0(*eKXO2E&!#;7$$"0/"[#\[Q*=!#9$"0(p(R+.#RV!#;7$$"/A'4y">5>!#8$"0%H#f-H;M$!#;7$$"0LZkXad#>!#9$"0k&zyY?3C!#;7$$"0>%ovC*>%>!#9$"0VR<en[X"!#;7$$"0j/j//p&>!#9$"0l%[=<g2g!#<7$$".-(*=))H(>!#7$!0Qu\![0ZH!#<7$$"0rDV4&p*)>!#9$!0()zA:+N>"!#;7$$"0i8R]QU+#!#9$!0#Rw')4iZ>!#;7$$"0G)p%yX*>?!#9$!0'3:4xMIF!#;7$$"0O2F%H<O?!#9$!0`c'41A-N!#;7$$"/C+^![?0#!#8$!01yPBT#>U!#;7$$"/2cM!3u1#!#8$!/Y^\]\v[!#:7$$"0p'\xEY%3#!#9$!0^Lpyv#fb!#;7$$"0d5F.(y*4#!#9$!0b'yYN8Kh!#;7$$"0^>G0\h6#!#9$!0n9<BO#*p'!#;7$$"0c7\Zv48#!#9$!0`0J;4E<(!#;7$$"0`J/n%=Z@!#9$!0Nj8Y/ak(!#;7$$"0t--EOC;#!#9$!0b3,Kxr/)!#;7$$"0[fgcy$y@!#9$!0Gk"G(=AU)!#;7$$"0lAS3lR>#!#9$!0RydV;Wu)!#;7$$"/o%Qs#G5A!#8$!0^UN;@Z.*!#;7$$"0%\R]&)*fA#!#9$!0h`'pn9p#*!#;7$$"0aFHDq?C#!#9$!0)>j2$4NY*!#;7$$"0v[O()3!eA!#9$!0&)*G[fe6'*!#;7$$"0Uq2"[lsA!#9$!0A<=v3"4(*!#;7$$"0-0vqS%*G#!#9$!0Z;3(4Xw(*!#;7$$"0d]O@aWI#!#9$!0r[zqouz*!#;7$$"0s,Euh/K#!#9$!0a@;:!G!y*!#;7$$"0lA.*HyNB!#9$!0T_-:"yE(*!#;7$$"0R3pE&z_B!#9$!0T>e.=li*!#;7$$"0#RS[m_nB!#9$!0&R<:&zj]*!#;7$$"/f;)*oC%Q#!#8$!/$ztH(HM$*!#:7$$"07t1f&[*R#!#9$!0t3_C=g9*!#;7$$"0u>aKahT#!#9$!0:Y]3xw!*)!#;7$$"0_1XiO0V#!#9$!0D?ZwUkn)!#;7$$"0Y_VQ;pW#!#9$!0nw3uZhQ)!#;7$$"0yFPUQFY#!#9$!07Rp)pW!3)!#;7$$"0bZp7]&yC!#9$!0p<+LO@v(!#;7$$"/RwhOI%\#!#8$!0n4rSmVS(!#;7$$"0i(*o(yV4D!#9$!0Em#>*>F0(!#;7$$"0lcUO)zDD!#9$!0.Y$=fMbm!#;7$$"0d&*\4"RTD!#9$!0%ptK!*)>E'!#;7$$"0`f)4<!yb#!#9$!0^Jw7uZ$e!#;7$$"0=$4oslsD!#9$!/>OUc@Qa!#:7$$"0VM4Wv!*e#!#9$!0xxF#R;"*\!#;7$$"0"ya*=,[g#!#9$!0jq_t8jb%!#;7$$"0#*[7m)[?E!#9$!0k-rExz6%!#;7$$"/JC'Guoj#!#8$!0;boA4tl$!#;7$$"0DT]/m>l#!#9$!0(yI@IAKK!#;7$$"07-/2Aum#!#9$!0Z))y-wxz#!#;7$$"0(o"*et[%o#!#9$!0dm=5)4@B!#;7$$"0;^)z&Q**p#!#9$!0G2v"G$R*=!#;7$$"0(*GuMSdr#!#9$!.^Q:\IY"!#97$$"0QI!Q."=t#!#9$!0)zff^hK5!#;7$$"0^9ly"eYF!#9$!0=1Og]<X'!#<7$$"0<jAnFBw#!#9$!0My'ewR@C!#<7$$"0(fI*p`zx#!#9$"0\.wqxUY"!#<7$$"0z$3"poYz#!#9$"0yd1#=y"[&!#<7$$"/kBZ*=%4G!#8$"0GWTkAu*))!#<7$$"0=@3oPk#G!#9$"0w!eb")fn7!#;7$$"0"HzvdzTG!#9$"/Jm%RDFf"!#:7$$"0%y*z!=*p&G!#9$"0D^ExF))*=!#;7$$"/()QCCLtG!#8$"0ktYQ9*4A!#;7$$"0@T>9R(*)G!#9$"/F^[_q-D!#:7$$"/nhDom/H!#8$"0LwS%za^F!#;7$$"0p%fi$>0#H!#9$"0-%o%[Rq*H!#;7$$"0BE$*))*)f$H!#9$"0*Gw[rh<K!#;7$$"0[9N5gF&H!#9$"0(4;Z-@NM!#;7$$"/tS,5FnH!#8$"0_h$y]C0O!#;7$$"0z$=&HPR)H!#9$"0;R$*4J&zP!#;7$$"0o4v[/'**H!#9$"03n#4.'G#R!#;7$$"0\V=N?^,$!#9$"0Mk!4hEXS!#;7$$"/L;f&[1.$!#8$"0Wpg%\U[T!#;7$$"0o0xM'[YI!#9$"0W*f7r!RB%!#;7$$"0`O#QTFjI!#9$"0E&zz%pII%!#;7$$"0>O#)*omyI!#9$"00%z$=PuM%!#;7$$"0M!))G:'Q4$!#9$"06Gn:ZPP%!#;7$$"01n*QV55J!#9$"0$3sO29$Q%!#;7$$"0'f()*y6j7$!#9$"0=+k"oytV!#;7$$"-X]EfTJ!#6$"0Ox,l0%[V!#;-%&COLORG6&%$RGBG$""!!""$"#5!""$""!!""-%*LINESTYLEG6#"""-%'CURVESG6%7hw7$$""!!""$"#5!""7$$"0P!fI`s^;!#;$"0YJ?olap*!#:7$$"0p4!4F)))3$!#;$"0e1NNk\Q*!#:7$$"0FS')[;^q%!#;$"0D+:'**z))*)!#:7$$"0*y*[rj?L'!#;$"0\)eDQ%Qa)!#:7$$"06')\My7&z!#;$"0#4S^VPf!)!#:7$$"08y4e(\_%*!#;$"0C*=#3<qd(!#:7$$"0h$e9Bp+6!#:$"0*[-cb*y/(!#:7$$"/T')G7Xh7!#9$"0b/LeUKZ'!#:7$$"0onse%p@9!#:$"04U@.nr(e!#:7$$"0B)zxX_'e"!#:$"0#)[UYwXC&!#:7$$"0OwJ#pqJ<!#:$"0al[3<\n%!#:7$$"0<Fx&y9&*=!#:$"/^3V8&R-%!#97$$"0Jek*)f#f?!#:$"0&z=>\_kL!#:7$$"02(*\D7u@#!#:$"0'Q4M/&ys#!#:7$$"0)p"yuH5O#!#:$"08cnOs?:#!#:7$$"0;hx$[!=`#!#:$"0C`2[)[u9!#:7$$"0J\&RPZwE!#:$"/)Ruv(f(4*!#:7$$"0$>Lk!eZ%G!#:$"0aNx0gKn#!#;7$$"0.Yk?=P*H!#:$!0T)=y1'G&G!#;7$$"0od74^r:$!#:$!/nH&eY-r)!#:7$$"0%*3h%yx7L!#:$!0MdFO_fS"!#:7$$"0cx%Q!e^Z$!#:$!09A(=!fv$>!#:7$$".#oWPFCO!#8$!/\![if'*R#!#97$$"0nb'y^6&y$!#:$!0:(fA0&z'G!#:7$$"0"G*[A%=_R!#:$!07xD5Q">L!#:7$$"0%=x?$=w4%!#:$!0DhdCk6o$!#:7$$"0X='G6paU!#:$!0pb07<(QS!#:7$$"/$4(3F'pT%!#9$!0*3m"zw/P%!#:7$$"0kf;z8dd%!#:$!0#f*3CLrl%!#:7$$"/FCFOJHZ!#9$!0>&*Gp3#)*[!#:7$$"0i-m0g)**[!#:$!0"*z">MnB^!#:7$$"0RT(3O5`]!#:$!0H6B$*o#)G&!#:7$$"/3$)4Qs;_!#9$!0%zbkKXCa!#:7$$"/8wI!))\O&!#9$!/A5%p4H^&!#97$$"08c$3S.Ya!#:$!0bXc^!RZb!#:7$$"0(4&f)*zq_&!#:$!0&e09**=sb!#:7$$"0)p![$zL.c!#:$!0txtY'y'e&!#:7$$"0*Hm$)efzc!#:$!0\O<A?If&!#:7$$"0rYHT2$fd!#:$!0Af4VC2f&!#:7$$"0VIA%*=!Re!#:$!0L!**>eczb!#:7$$"0KY?`^p"f!#:$!0aJ;RR-c&!#:7$$"0@i=7%)[*f!#:$!02&)pL]F`&!#:7$$"0j.,_g!eh!#:$!0vF?`4&\a!#:7$$"04&e&y=_J'!#:$!/#)y[$zxL&!#97$$"0664"e$fZ'!#:$!0PHPIEK>&!#:7$$"0;B"=?KNm!#:$!0s***pDZ@]!#:7$$"0"*R$*Q"y"y'!#:$!0*3^L+TS[!#:7$$"0(eoc.k\p!#:$!0#o$=D_wg%!#:7$$"0PT"=ax*4(!#:$!0&flMIgyV!#:7$$"0"Hl22&)fs!#:$!0c]G-j[6%!#:7$$"0<i[=jIT(!#:$!0*)\22\c%Q!#:7$$"0c>2&f=$e(!#:$!0:7CAO+`$!#:7$$"0)[nl(*\Ix!#:$!0OZBw#oWK!#:7$$"0p%HjAq(*y!#:$!0ugkr?(4H!#:7$$"0$pO)=*3]!)!#:$!09q$RoP'f#!#:7$$"06Le`wn@)!#:$!0_u$>`IZA!#:7$$"0#4qE&*fg$)!#:$!0C5-bhF%>!#:7$$"0Cg^7(RC&)!#:$!0c!=U3N%f"!#:7$$"0^8*=vh#o)!#:$!0@$faVNe7!#:7$$"0>66bM2%))!#:$!0'yC[78_#*!#;7$$"0ru#**)p#)**)!#:$!0UY!)GC*yf!#;7$$"0$=h]?h\"*!#:$!0w?@mi]*G!#;7$$"0A-U#p@8$*!#:$"06Z.I(QLN!#<7$$"0O"fJU9p%*!#:$"0p,\.\+N$!#;7$$"0(4B!Q]Kj*!#:$"0)HX6_V#Q'!#;7$$"0[nD'f!=y*!#:$"0\0pX\m+*!#;7$$"0-!)4p()f%**!#:$"0^Pt3gf<"!#:7$$"0Q6x^C.,"!#9$"0\a'fE%RU"!#:7$$"0\7%*)>,E5!#9$"/>qYQ&\l"!#97$$"0n1Wh(RU5!#9$"09W+k2y(=!#:7$$"0"[?t$*[d5!#9$"0kvG)*\c1#!#:7$$"0pl&)RXH2"!#9$"0Fiyb2,C#!#:7$$"0W!3(o5+4"!#9$"0*>`oU46C!#:7$$"0t9!3>Y06!#9$"0YsJKRfa#!#:7$$"0`#fvOE@6!#9$"0'=B2B(Rm#!#:7$$"0%R>mOLP6!#9$"0Tf$z:HjF!#:7$$"02yY60@:"!#9$"/"**=6wh$G!#97$$"0tE/+^y;"!#9$"0o!y.pe%*G!#:7$$"0apu-xM="!#9$"0eZN1KJ$H!#:7$$"0OZ#>?>+7!#9$"0U]2RfL&H!#:7$$"0'**RvA%\@"!#9$"0))z^FAO&H!#:7$$"0v%)*35'>B"!#9$"0&[^t$)3MH!#:7$$"0[cR5>tC"!#9$"0x]P;$*))*G!#:7$$"0Thh8:DE"!#9$"0R<E!\Q[G!#:7$$"0E_Dvb)y7!#9$"0a?_6Mvx#!#:7$$"0y/,Zi_H"!#9$"/,"e#3;!p#!#97$$"0F!y`,>58!#9$"0mKbWhuf#!#:7$$"0Ee2pUgK"!#9$"0.8)=UE'[#!#:7$$"/)*[<K^T8!#8$"0ujXH;hO#!#:7$$"0/y;V$Ge8!#9$"0B9^WWTA#!#:7$$"0(3dHVzs8!#9$"0RX^?(\#4#!#:7$$"0OZLig%*Q"!#9$"0my$3BZK>!#:7$$"0Ctc"y709!#9$"0_@!)\ZYx"!#:7$$"012+oV1U"!#9$"00W&HGZ7;!#:7$$"0(oK()=<O9!#9$"0^()*=2ZX9!#:7$$"0Dpen4?X"!#9$"0ws'[Q[r7!#:7$$"/,Smuzo9!#8$"0Y5jbdV3"!#:7$$"0w*RE->%["!#9$"09,%f96:"*!#;7$$"0"R/d[Q*\"!#9$"0kriV"f2u!#;7$$"0jIrmFc^"!#9$"0cVtx??f&!#;7$$"0`R!=^$=`"!#9$"0B\Nx:<!Q!#;7$$"0P,xE#QY:!#9$"0Z$[$)z+AA!#;7$$"00;bT9Oc"!#9$"0a+oq<y%R!#<7$$"0Z4xZL!y:!#9$!0,*o`ZH*3"!#;7$$"0/J0>)*[f"!#9$!09[&[(pRw#!#;7$$"0MU^#R*4h"!#9$!08%)zK8CH%!#;7$$"0`E*)\l`i"!#9$!0_I"Gp3$f&!#;7$$"0$GsOy_T;!#9$!/TnM8Bxp!#:7$$"0T&)*3tzd;!#9$!0%3$G/_.G)!#;7$$"0H'Gb%*)Rn"!#9$!02tM?>=[*!#;7$$"0@Ywk,!*o"!#9$!0D%\bik]5!#:7$$"0z%H.fa/<!#9$!06i6)*Rt9"!#:7$$"0%Gs%z@1s"!#9$!0^bf&o(pB"!#:7$$"/KcIhkO<!#8$!0WAkG,cJ"!#:7$$"0D;'H"HJv"!#9$!0KIP5K^Q"!#:7$$"02aTOZww"!#9$!0vTA%)onV"!#:7$$"0:4wX"*Ry"!#9$!0)Q#*)[;T["!#:7$$"0E#[hES+=!#9$!/QlB3A?:!#97$$"09Ot*y@;=!#9$!0%H*e:*GW:!#:7$$"08=mkz0$=!#9$!0#G[Gb<d:!#:7$$"0b7c:dw%=!#9$!0KMc(zjh:!#:7$$"0O"zXS7i=!#9$!0wiEqUkb"!#:7$$"0jp#yC&*y=!#9$!00:XV2/a"!#:7$$"0/"[#\[Q*=!#9$!0z(=(31w^"!#:7$$"/A'4y">5>!#8$!01qs\QP["!#:7$$"0LZkXad#>!#9$!0()>@%\MV9!#:7$$"0>%ovC*>%>!#9$!0"4a;TE$R"!#:7$$"0j/j//p&>!#9$!0H"yuGjS8!#:7$$".-(*=))H(>!#7$!0U3&y(*Hx7!#:7$$"0rDV4&p*)>!#9$!01!*z=b\?"!#:7$$"0i8R]QU+#!#9$!0OD'*f"4P6!#:7$$"0G)p%yX*>?!#9$!0#Rb-dFf5!#:7$$"0O2F%H<O?!#9$!/u<7e)fu*!#:7$$"/C+^![?0#!#8$!//2d^-#)))!#:7$$"/2cM!3u1#!#8$!0J%y^Zz=!)!#;7$$"0p'\xEY%3#!#9$!0=Laysg.(!#;7$$"0d5F.(y*4#!#9$!0oh^!)>w8'!#;7$$"0^>G0\h6#!#9$!06#Gu6&)o^!#;7$$"0c7\Zv48#!#9$!0kLYx#3)G%!#;7$$"0`J/n%=Z@!#9$!0rMuEezK$!#;7$$"0t--EOC;#!#9$!0Y!RQSeKC!#;7$$"0[fgcy$y@!#9$!0ar5J`/^"!#;7$$"0lAS3lR>#!#9$!0L=X&)3iF'!#<7$$"/o%Qs#G5A!#8$"0:iuwXdr#!#<7$$"0%\R]&)*fA#!#9$"0#e1(Gr'36!#;7$$"0aFHDq?C#!#9$"0W7OLm3$>!#;7$$"0v[O()3!eA!#9$"0$[d!oU$3F!#;7$$"0Uq2"[lsA!#9$"0O+=OpiQ$!#;7$$"0-0vqS%*G#!#9$"0#R`]_"o6%!#;7$$"0d]O@aWI#!#9$"07d]w"\DZ!#;7$$"0s,Euh/K#!#9$"0e>&)e6aK&!#;7$$"0lA.*HyNB!#9$"0ymz'H=]e!#;7$$"0R3pE&z_B!#9$"0=')G7iUP'!#;7$$"0#RS[m_nB!#9$"0>%[pj(px'!#;7$$"/f;)*oC%Q#!#8$"0Pu+aOb<(!#;7$$"07t1f&[*R#!#9$"/tOF2+%[(!#:7$$"0u>aKahT#!#9$"0%*H<qG8w(!#;7$$"0_1XiO0V#!#9$"0/$Q,'p-&z!#;7$$"0Y_VQ;pW#!#9$"0\&zbE64")!#;7$$"0yFPUQFY#!#9$"0*[ikhP1#)!#;7$$"0bZp7]&yC!#9$"/A\a\b\#)!#:7$$"/RwhOI%\#!#8$"/7.9BFS#)!#:7$$"0i(*o(yV4D!#9$"0%4N"4WP=)!#;7$$"0lcUO)zDD!#9$"0KRMqi@2)!#;7$$"0d&*\4"RTD!#9$"0NUI(e=>z!#;7$$"0`f)4<!yb#!#9$"0Ghpt@;r(!#;7$$"0=$4oslsD!#9$"0&4-e/&\[(!#;7$$"0VM4Wv!*e#!#9$"0`CObfW>(!#;7$$"0"ya*=,[g#!#9$"0&\*4h@)zo!#;7$$"0#*[7m)[?E!#9$"0:&e*\HN`'!#;7$$"/JC'Guoj#!#8$"0sV#eSjSh!#;7$$"0DT]/m>l#!#9$"0E3yCkOv&!#;7$$"07-/2Aum#!#9$"03hE%*GbL&!#;7$$"0(o"*et[%o#!#9$"08.VZo>&[!#;7$$"0;^)z&Q**p#!#9$"0p*oz%zxR%!#;7$$"0(*GuMSdr#!#9$"0lf4<81#R!#;7$$"0QI!Q."=t#!#9$"0@_<nVcU$!#;7$$"0^9ly"eYF!#9$"0Y%yCW,lH!#;7$$"0<jAnFBw#!#9$"0w\j.26Z#!#;7$$"0(fI*p`zx#!#9$"0w]]=Q5)>!#;7$$"0z$3"poYz#!#9$"0Jg+qv-Y"!#;7$$"/kBZ*=%4G!#8$"0L5bDCk+"!#;7$$"0=@3oPk#G!#9$"0-5Ny*RD\!#<7$$"0"HzvdzTG!#9$"0;UV6vK/%!#=7$$"0%y*z!=*p&G!#9$!0,,G&\%o$R!#<7$$"/()QCCLtG!#8$!0iOp"*>UV)!#<7$$"0@T>9R(*)G!#9$!0Qmy(3![F"!#;7$$"/nhDom/H!#8$!0kP7kVyk"!#;7$$"0p%fi$>0#H!#9$!0>L>qa>-#!#;7$$"0BE$*))*)f$H!#9$!02>XILOO#!#;7$$"0[9N5gF&H!#9$!0s2BrPjq#!#;7$$"/tS,5FnH!#8$!/LDa'\&yH!#:7$$"0z$=&HPR)H!#9$!0:8t'GHiK!#;7$$"0o4v[/'**H!#9$!0$[?cb/+N!#;7$$"0\V=N?^,$!#9$!0cVc8Lsq$!#;7$$"/L;f&[1.$!#8$!/#*GS"Qg)Q!#:7$$"0o0xM'[YI!#9$!0v0z&[yQS!#;7$$"0`O#QTFjI!#9$!/$y<w:!oT!#:7$$"0>O#)*omyI!#9$!0V-KpDrD%!#;7$$"0M!))G:'Q4$!#9$!0_g,fXyJ%!#;7$$"01n*QV55J!#9$!0(H)H/\LN%!#;7$$"0'f()*y6j7$!#9$!0ZYIx6#fV!#;7$$"-X]EfTJ!#6$!08#)*HE]QV!#;-%&COLORG6&%$RGBG$"#5!""$""!!""$"#5!""-%*LINESTYLEG6#"""-%%VIEWG6$;$""!!""$"+aEfTJ!"*%(DEFAULTG-&%&_AXISG6#"""6#-%+_TICKMARKSG6#%)_PITICKSG-%+AXESLABELSG6$-I#miG6#/I+modulenameG6"I,TypesettingGI(_syslibG6"65Q"x6"/%'familyGQ!6"/%%sizeGQ#106"/%%boldGQ&false6"/%'italicGQ%true6"/%*underlineGQ&false6"/%*subscriptGQ&false6"/%,superscriptGQ&false6"/%+foregroundGQ([0,0,0]6"/%+backgroundGQ.[255,255,255]6"/%'opaqueGQ&false6"/%+executableGQ&false6"/%)readonlyGQ&false6"/%)composedGQ&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%ROOTG6'-%)BOUNDS_XG6#$"$?&!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"%+M!""-%.BOUNDS_HEIGHTG6#$"%?Q!""-%)CHILDRENG6"

Figura 1 Autovalores 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 est\303\243o em preto s\303\263lido, azul, vermelho, verde, e magenta, respectivamente, enquanto LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYkLUkjbW5HRiQ2JFEiMEYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R0Y9RjU= est\303\241 em preto pontilhado

Uma vez que a equa\303\247\303\243o diferencial n\303\243o \303\251 auto-adjunta, n\303\243o estamos seguros de autofun\303\247\303\265es ortogonais. Os produtos internos 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s\303\243o computados e mostrados na Tabela 1. Se essas autofun\303\247\303\265es forem ortononais, as entradas da diagonal secund\303\241ria deveriam ser zero. Que elas n\303\243o sejam \303\251 nossa evid\303\252ncia que as autofun\303\247\303\265es n\303\243o sejam ortogonais.

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiUiIidGKWYqNiRJImlHRidJImpHRidGJzYkSSlvcGVyYXRvckdGJ0kmYXJyb3dHRidGJy1JJGludEdGJDYkKiYmSSRwaGlHRic2IzkkIiIiJkY2NiM5JUY5L0kieEdGJzsiIiFJI1BpR0YlRidGJ0Yn

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKk8oKnp6JQ==

Tabela 1 Os produtos internos 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

Gastamos bastante energia separando partes real e imagin\303\241ria na solu\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==, mais em mostrando que essas partes s\303\243o m\303\272ltiplos de si. Um modo mais direto de encontrar a forma trigonom\303\251trica simples que n\303\263s usamos para gerar as autofun\303\247\303\265es \303\251 obter a solu\303\247\303\243o geral da equa\303\247\303\243o diferencial sob a hip\303\263tese que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjFGJ0YyRjI=. Isso leva a

QyQ+SSV0ZW1wRzYiLUkpYXNzdW1pbmdHSShfc3lzbGliR0YlNiQ3Iy1JJ2Rzb2x2ZUc2JCUqcHJvdGVjdGVkR0YoNiRJJGRlcUdGJS1JInlHRiU2I0kieEdGJTcjLUkiPkdGLjYkSSdsYW1iZGFHRiUiIiIhIiI= QyQ+STcjbXN1YihtaSgiWSIpLG1uKCIzIikpRzYiLUkodW5hcHBseUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLUklZXZhbEdGKTYkLUkkcmhzR0YpNiNJJXRlbXBHRiU3JC9JJF9DMUdGJSZJInJHRiU2IyIiJC9JJF9DMkdGJSZJInNHRiVGOEkieEdGJSEiIg== Ly0uSTcjbXN1YihtaSgiWSIpLG1uKCIzIikpRzYiNiNJInhHRiYtRiVGJw==

Ly1JNyNtc3ViKG1pKCJZIiksbW4oIjMiKSlHNiI2I0kieEdGJSwmKigmSSJyR0YlNiMiIiQiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRichIiJGLi1JJHNpbkdGMTYjKiYsJkY2Ri5JJ2xhbWJkYUdGJUYuI0YuIiIjRidGLkYuRi4qKCZJInNHRiVGLEYuRi9GLi1JJGNvc0dGMUY5Ri5GLg==

e, depois de aplicar as condi\303\247\303\265es limites, para

PyhJImtHNiIiIiJGJSIiI0kldHJ1ZUclKnByb3RlY3RlZEc+JkkicUdGJDYjLCYiIiVGJUYjRiUtSSVldmFsR0YoNiQmSSNiY0dGJDYjRiMvSSJ5R0YkSTcjbXN1YihtaSgiWSIpLG1uKCIzIikpR0Yk

LywmKiYmSSJyRzYiNiMiIiQiIiIsJiEiIkYqSSdsYW1iZGFHRidGKiNGKiIiI0YpJkkic0dGJ0YoRi8iIiE=

LywqKigmSSJyRzYiNiMiIiQiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2IywkSSNQaUdGLiEiIkYqLUkkc2luR0YtNiMqJiwmRjNGKkknbGFtYmRhR0YnRiojRioiIiNGMkYqRipGMyoqRiVGKkYrRiotSSRjb3NHRi1GNkYqRjhGOkY7KigmSSJzR0YnRihGKkYrRipGPUYqRjMqKkZARipGK0YqRjRGKkY4RjohIiMiIiE=

Se agora expressarmos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, digamos, em termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==, como em

LUkoaXNvbGF0ZUc2IjYkJkkicUdGJDYjIiImJkkickdGJDYjIiIk

LyZJInJHNiI2IyIiJCwkKiYmSSJzR0YlRiYiIiIsJiEiIkYsSSdsYW1iZGFHRiVGLCNGLiIiIyMhIiNGJw==

podemos escrever essa nova solu\303\247\303\243o geral como

PkkjWWRHNiIqJiZJInNHRiQ2IyIiJCEiIi1JKGNvbGxlY3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1JJWV2YWxHRi42JC1JNyNtc3ViKG1pKCJZIiksbW4oIjMiKSlHRiQ2I0kieEdGJC8mSSJyR0YkRigsJComRiYiIiIsJkYqRj1JJ2xhbWJkYUdGJEY9I0YqIiIjIyEiI0YpNyRJJGV4cEdGLUYmRj0=

KiYsJiomLCYhIiIiIiJJJ2xhbWJkYUc2IkYnI0YmIiIjLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YpNiMqJkYlI0YnRitJInhHRilGJ0YnIyEiIyIiJC1JJGNvc0dGLkYxRidGJy1JJGV4cEdGLjYjLCRGNEYmRic=

Para mostrar que isso \303\251 equivalente ao nosso LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImNGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1Rj1GPUY9, mostramos que a diferen\303\247a 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 desaparece.

LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywmSSNZZEdGJyIiIkkjWWNHRichIiI=

IiIh

JSFH