<Text-field style="Heading 1" layout="Heading 1">Introdu\303\247\303\243o</Text-field>

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Coeficientes a Determinar

A equa\303\247\303\243o diferencial ordin\303\241ria de coeficientes constantes \303\251 uma passagem padr\303\243o para um primeiro curso de equa\303\247\303\265es diferencias. A solu\303\247\303\243o da equa\303\247\303\243o n\303\243o-homog\303\252nea \303\251 uma soma de solu\303\247\303\265es gerais da equa\303\247\303\243o homog\303\252nea, mais uma solu\303\247\303\243o particular da equa\303\247\303\243o n\303\243o-homog\303\252nea.

A solu\303\247\303\243o parcial \303\251 geralmente encontrada por qualquer varia\303\247\303\243o de par\303\242metros, ou coeficientes indeterminados. Varia\303\247\303\243o de par\303\242metros \303\251 aplic\303\241vel em todos os casos, mas as integra\303\247\303\265es que ela necessita podem ser formid\303\241veis, mesmo para um sistema computacional de \303\241lgebra. Por outro lado, o m\303\251todo dos coeficientes indeterminados come\303\247a com c\303\241lculos bem simples, mas \303\251 v\303\241lido para uma classe restrita de problemas.

As manipula\303\247\303\265es no m\303\251todo dos coeficientes indeterminados come\303\247a com uma solu\303\247\303\243o teste, uma "adivinha\303\247\303\243o" na "forma" de uma solu\303\247\303\243o parcial, que \303\251 apenas uma combina\303\247\303\243o linear de um conjunto de fun\303\247\303\265es determinado quase inteiramente pelo termo n\303\243o-homog\303\252neo. Embora os c\303\241lculos seguros sejam meramente uma substitui\303\247\303\243o dessa tentativa de solu\303\247\303\243o de volta na equa\303\247\303\243o diferencial, esse m\303\251todo \303\251 dif\303\255cil para estudantes usarem por causa das regras para determinar a tentativa de solu\303\247\303\243o n\303\243o serem transparentes.

Essas regras podem ser reduzidas para um algoritmo, mas sem algum conhecimento da base para o algoritmo, os estudantes parecem encontrar um t\303\263pico inteiramente opaco. Atrav\303\251s desses v\303\241rios anos ensinando cursos em equa\303\247\303\265es diferenciais, tentei uma variedade de dispositivos para tornar mais f\303\241cil para estudantes compreender a conex\303\243o entre o termo n\303\243o-homog\303\252neo e o chute inicial para o m\303\251todo.

A vers\303\243o mais recente da minha aproxima\303\247\303\243o come\303\247a com a ressalva que o m\303\251todo se aplica somente se o termo n\303\243o-homog\303\252neo possui ele mesmo um "fragmento" de uma solu\303\247\303\243o geral de alguma EDO linear homog\303\252nea de coeficientes constantes. A solu\303\247\303\243o completa que cont\303\251m esse fragmento foi gerada por um conjunto de ra\303\255zes caracter\303\255sticas. O "truque" para os estudantes era determinar quais das ra\303\255zes caracter\303\255sticas (latentes) escondidas eles poderiam possuir.

Por um longo tempo eu pensei que os exerc\303\255cios de estudantes em um processo "avan\303\247ado" de escrever uma solu\303\247\303\243o geral a partir de um conjunto de ra\303\255zes caracter\303\255sticas levaria a uma compreens\303\243o essencial. Al\303\251m do mais, descobri que o estudante t\303\255pico continua tendo dificuldade pensando de modo "retr\303\263grado", do termo n\303\243o-homog\303\252neo para as ra\303\255zes caracter\303\255sticas (ou latentes) que o geraram.

Por um tempo igualmente longo depois, eu forneci a alunos uma tabela com exemplos onde amostras do lado direito, ra\303\255zes latentes, e as tentativas de solu\303\247\303\243o consequentes est\303\243o listadas.

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Ra\303\255zes Latentes

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(10) 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY6LUkjbWlHRiQ2I1EhRictSSNtb0dGJDY+USwmUGx1c01pbnVzO0YnLyUnZmFtaWx5R1Emc2VyaWZGJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjsvJSp1bmRlcmxpbmVHRjsvJSpzdWJzY3JpcHRHRjsvJSxzdXBlcnNjcmlwdEdGOy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnLyUnb3BhcXVlR0Y7LyUrZXhlY3V0YWJsZUdGOy8lKXJlYWRvbmx5R0Y7LyUpY29tcG9zZWRHRjsvJSpjb252ZXJ0ZWRHRjsvJStpbXNlbGVjdGVkR0Y7LyUscGxhY2Vob2xkZXJHRjsvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjsvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMDU1NTU1NmVtRictRiw2NVEiaUYnRjNGNkY5L0Y9USV0cnVlRidGPkZARkJGREZHRkpGTEZORlBGUkZURlZGWC9GZW5RJ2l0YWxpY0YnLUYwNj5RIixGJ0YzRjZGOUY8Rj5GQEZCRkRGR0ZKRkxGTkZQRlJGVEZWRlhGWkZnbkZpbkZbb0Zdb0Zfb0Zhb0Zjby9GZm9RLDAuMjc3Nzc3OGVtRicvRmlvRmZwLUYwNj5GMkYzRjZGOUY8Rj5GQEZCRkRGR0ZKRkxGTkZQRlJGVEZWRlhGWkZnbkZpbkZbb0Zdb0Zfb0Zhb0Zjby9GZm9RLDAuMjIyMjIyMmVtRicvRmlvRltxRltwRjNGNkY5RjxGPkZARkJGREZHRkpGTEZORlBGUkZURlZGWEZa

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

Tabela 5.10.2 Exemplos de termos n\303\243o-homog\303\252neos e da forma de 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 que eles requerem

Infelizmente, essa aproxima\303\247\303\243o n\303\243o explica nem porque o m\303\251todo de coeficientes indeterminados funciona para termos n\303\243o-homog\303\252neos dos tipos listados na Tabela 5.10.2, tampouco porque uma combina\303\247\303\243o linear dos termos gerados pelas ra\303\255zes latentes deveriam ser uma tentativa de solu\303\247\303\243o.

Por outro lado, eu aprendi o m\303\251todo dos coeficientes indeterminados de uma antiga edi\303\247\303\243o do Rainville's Elementary Differential Equations (The Macmillan Company, 1958). O tratamento de Rainville come\303\247a com uma discuss\303\243o (te\303\263rica) simb\303\263lica do operador de aniquila\303\247\303\243o que poderia converter o lado esquerdo da equa\303\247\303\243o para uma equa\303\247\303\243o de ordem maior, e reduzir o lado direito a zero. A solu\303\247\303\243o geral da EDO de nova ordem foi ent\303\243o uma soma da solu\303\247\303\243o homog\303\252nea da equa\303\247\303\243o original mais o "chute" para a solu\303\247\303\243o particular necess\303\241ria no m\303\251todo dos coeficientes indeterminados.

Por v\303\241rios anos procurei por uma abordagem do Maple ao longo dos anos de meu conhecimento de Rainville. Eu procurei ferramentas para gerar e aplicar automaticamente o operador de aniquila\303\247\303\243o, ilustrando assim a teoria que fez o m\303\251todo transparente para mim quando primeiramente o aprendi.

Os c\303\241lculos nesse artigo demonstram como as seis ferramentas de software listadas na Tabela 1 tornaram poss\303\255vel ensinar o m\303\251todo dos coeficientes indeterminados da maneira que aprendi. O c\303\263digo que cria essas ferramentas est\303\241 escondido no bloco colapsado de documentos na seguinte se\303\247\303\243o Initializations. O bloqueio pode ser aberto colocando o cursor na sa\303\255da de cor azul ("Maple initializations complete") e usando o menu View para selecionar Expand Document Block.

Comando

Coment\303\241rio

AnnihilatingOperator

Cria um operador de aniquila\303\247\303\243o

displayop

Mostra um aniquilador como um operador diferente

ApplyOperator

Aplica um aniquilador em ambos os lados de uma ED

ExtractCharacteristicRoots

Lista as ra\303\255zes caracter\303\255sticas para uma ED homog\303\252nea

ExtractLatentRoots

Lista as ra\303\255zes latentes para um termo n\303\243o-homog\303\252neo

FromLargeSolutionDeleteSmall

Extrai uma solu\303\247\303\243o teste de solu\303\247\303\265es homog\303\252neas da ED aniquilada

Tabela 1 Seis comandos do Maple para ensinar coeficientes indeterminados

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

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STtFeHRyYWN0Q2hhcmFjdGVyaXN0aWNSb290c0dGKGYqNiNJIlhHRig2I0kiTEdGKEYoRihDJD44JC0mSShERXRvb2xzR0YlNiNJKmRlMmRpZmZvcEdGKDYlLUkkbGhzR0YmNiM5JC1JInlHRig2I0kidEdGKDckSSNEdEdGKEZDLUkmc29sdmVHRig2JEY1RkVGKEYoRig3I0Yu

Maple initializations completed

<Text-field style="Heading 1" layout="Heading 1">Coeficientes Indeterminados</Text-field>

Para ilustrar o que um estudante vivencia quando se depara com uma equa\303\247\303\243o diferencial cuja solu\303\247\303\243o particular deve ser obtida pelo m\303\251todo dos coeficientes indeterminados, considere a equa\303\247\303\243o diferencial

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O primeiro passo seria a determina\303\247\303\243o das ra\303\255zes latentes 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 pela inspe\303\247\303\243o do termo n\303\243o-homog\303\252neo 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. O pr\303\263ximo passo seria a determina\303\247\303\243o das ra\303\255zes caracter\303\255sticas para a equa\303\247\303\243o diferencial homog\303\252nea

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No Maple, isso pode ser feito obtendo a equa\303\247\303\243o caracter\303\255stica decorrente do "chute exponencial"

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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNFR0dGKC1JJGV4cEdGJTYjKiZJJ2xhbWJkYUdGKCIiIkkidEdGKEYzNyNGLg==

A substitui\303\247\303\243o leva a

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O uso do pric\303\255pio zero para n\303\272meros reais ou complexos leva ent\303\243o \303\240 equa\303\247\303\265es caracter\303\255sticas

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\303\211 claro, o estudante experiente poderia escrever essa equa\303\247\303\243o por inspe\303\247\303\243o. Em qualquer evento, a solu\303\247\303\243o da equa\303\247\303\243o caracter\303\255stica d\303\241

LUkmc29sdmVHNiI2JEkjQ0VHRiRJJ2xhbWJkYUdGJA==

NiReJCEiIiIiJF4kRiQhIiQ=

para as ra\303\255zes caracter\303\255sticas. Desse modo, as ra\303\255zes latentes repetem as ra\303\255zes caracter\303\255sticas, e ent\303\243o a solu\303\247\303\243o teste apropriada para coeficientes indeterminados poderia ser

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A determina\303\247\303\243o dos coeficientes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= segue de uma substitui\303\247\303\243o direta na equa\303\247\303\243o diferencial, um passo resultando na equa\303\247\303\243o

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Mas essa "equa\303\247\303\243o" deve ser uma identidade em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, em cujo caso o comando do Maple

PkkjcTJHNiItSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYkLUkpaWRlbnRpdHlHRiQ2JEkjcTFHRiRJInRHRiQ8JEkiYUdGJEkiYkdGJA==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxMkdGKDwkL0kiYUdGKCMhIiIiIicvSSJiR0YoIiIhNyNGLg==

produz os valores apropriados dos coeficientes desconhecidos. \303\211 claro, trabalhando manualmente sem qualquer tecnologia requer coeficientes correspondentes de 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 e 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 in

PkkjcTNHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSNxMUdGJA==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxM0dGKC8sJComLUkkZXhwR0YlNiMsJEkidEdGKCEiIiIiIiwmKiZJImFHRihGNy1JJHNpbkdGJTYjLCRGNSIiJEY3RjYqJkkiYkdGKEY3LUkkY29zR0YlRj1GN0Y3RjciIicqJkYxRjdGO0Y3NyNGLg==

O Maple acha isso mais f\303\241cil de fazer se o fator LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtRiw2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y5USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUYsNiVRInRGJ0Y1RjhGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGK0ZB n\303\243o estivesse presente nos dois lados. Portanto, somos levados a

PkkjcTRHNiItSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJI3EzR0YkIiIiLUkkZXhwR0YnNiMsJEkidEdGJCEiIkYz

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNxNEdGKC8sJiomSSJhR0YoIiIiLUkkc2luR0YlNiMsJEkidEdGKCIiJEYyISInKiZJImJHRihGMi1JJGNvc0dGJUY1RjIiIidGMzcjRi4=

e ent\303\243o \303\240s equa\303\247\303\265es

QyQ+SSRlcTFHNiItSSRtYXBHJSpwcm90ZWN0ZWRHNiVJJmNvZWZmR0YoSSNxNEdGJS1JJGNvc0c2JEYoSShfc3lzbGliR0YlNiMsJEkidEdGJSIiJCIiIg==PkkkZXEyRzYiLUkkbWFwRyUqcHJvdGVjdGVkRzYlSSZjb2VmZkdGJ0kjcTRHRiQtSSRzaW5HNiRGJ0koX3N5c2xpYkdGJDYjLCRJInRHRiQiIiQ=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigvLCRJImJHRigiIiciIiE3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvLCRJImFHRighIiciIiI3I0Yu

Deveria ser claro que a solu\303\247\303\243o dessas duas equa\303\247\303\265es tamb\303\251m produz a solu\303\247\303\243o particular

LyYtSSJ5RzYiNiNJInRHRiY2I0kicEdGJi1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJI1lUR0YmSSNxMkdGJg==

LyYtSSJ5RzYiNiNJInRHRiY2I0kicEdGJiwkKihGKCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLCRGKCEiIkYtLUkkY29zR0YwNiMsJEYoIiIkRi0jRjUiIic=

<Text-field style="Heading 1" layout="Heading 1">Usando as Ferramentas</Text-field>

Para ilustrar como o conjunto de ferramentas listados na Tabela 1 funcionam, consideremos de novo a equa\303\247\303\243o diferencial

SSNERUc2Ig==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCgtSSVkaWZmR0YmNiQtSSJ5R0YoNiNJInRHRigtSSIkR0YmNiRGNCIiIyIiIi1GLzYkRjFGNEY4RjEiIzUqJi1JJGV4cEdGJTYjLCRGNCEiIkY5LUkkc2luR0YlNiMsJEY0IiIkRjk3Iy8sKC1GLzYkRjpGNEY5RjpGOEYxRjxGPQ==

O aniquilador do lado direito da equa\303\247\303\243o diferencial \303\251 obtido com o comando AnnihilatingOperator, mas mostrado como um operador com o comando displayop. Juntos o resultado \303\251

QyQ+SSNSS0c2Ii1JNUFubmloaWxhdGluZ09wZXJhdG9yR0YlNiMtSSRyaHNHJSpwcm90ZWN0ZWRHNiNJI0RFR0YlIiIiLUkqZGlzcGxheW9wRzYiNiNJI1JLR0Yk

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNSS0dGKCwoKiRJI0R0R0YoIiIjIiIiRjBGMSIjNUYyNyNGLg==

LCgtSSZtZnJhY0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nRzYkRiZJKF9zeXNsaWJHRik2JC1JJW1zdXBHRiU2JC1JI21pR0YlNiNRImRGKS1GMjYjUSIyRiktRi82JC1GMjYjUSNkdEYpRjUiIiItRiQ2JEYxRjoiIiMiIzVGPQ==

Para verificar que esse operador realmente aniquila o lado direito da equa\303\247\303\243o diferencial, precisamos aplicar o operador a aquele lado direito. Isso \303\251 feito com o comando ApplyOperator, e resulta em

LUkuQXBwbHlPcGVyYXRvckc2IjYkSSNSS0dGJC1JJHJoc0clKnByb3RlY3RlZEc2I0kjREVHRiQ=

IiIh

O operador de aniquila\303\247\303\243o precisa agora ser aplicado \303\240 equa\303\247\303\243o diferencial completa. O lado direito ser\303\241 aniquilado, e o lado esquerdo se tornar\303\241 uma equa\303\247\303\243o linear de coeficiente constante de ordem maior. Novamente usando o comando ApplyOperator, temos

PkkmbmV3ZGVHNiItSS5BcHBseU9wZXJhdG9yR0YkNiRJI1JLR0YkSSNERUdGJA==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZuZXdkZUdGKC8sLC1JJWRpZmZHRiY2JC1JInlHRig2I0kidEdGKC1JIiRHRiY2JEY2IiIlIiIiLUYxNiRGMy1GODYkRjYiIiRGOi1GMTYkRjMtRjg2JEY2IiIjIiNDLUYxNiRGM0Y2IiNTRjMiJCsiIiIhNyMvLCwtRjE2JC1GMTYkLUYxNiRGR0Y2RjZGNkY7RlFGOkZTRkZGR0ZJRjNGSkZL

Essa nova equa\303\247\303\243o diferencial dita o chute apropriado na forma da solu\303\247\303\243o particular. Uma maneira de encontrar a conex\303\243o \303\251 atrav\303\251s das ra\303\255zes caracter\303\255sticas, encontrada com o comando ExtractCharacteristicRoots.

LUk7RXh0cmFjdENoYXJhY3RlcmlzdGljUm9vdHNHNiI2I0kmbmV3ZGVHRiQ=

NiZeJCEiIiEiJF4kRiQiIiRGI0Ym

As ra\303\255zes caracter\303\255sticas para a equa\303\247\303\243o homog\303\252nea determinadas pela equa\303\247\303\243o diferencial original s\303\243o

LUk7RXh0cmFjdENoYXJhY3RlcmlzdGljUm9vdHNHNiI2Iy9JJEhERUdGJCIiIQ==

NiReJCEiIiIiJF4kRiQhIiQ=

Dessa forma, a equa\303\247\303\243o homog\303\252nea horiginal possui ra\303\255zes caracter\303\255sticas 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, mas a nova equa\303\247\303\243o possui ra\303\255zes caracter\303\255sticas 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, e de novo 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 O segundo par de ra\303\255zes complexas s\303\243o as "ra\303\255zes latentes", as ra\303\255zes se escondendo no termo n\303\243o-homog\303\252neo da equa\303\247\303\243o diferencial original. Essas s\303\243o as ra\303\255zes que os estudantes acham t\303\243o dif\303\255cil extrair por simplesmente inspecionar o termo n\303\243o-homog\303\252neo.

As ra\303\255zes latentes podem ser encontradas com o comando ExtractLatentRoots, como vemos de

LUkzRXh0cmFjdExhdGVudFJvb3RzRzYiNiNJI0RFR0Yk

NiReJCEiIiIiJF4kRiQhIiQ=

Entretanto, estudantes poderiam continuar a n\303\243o entender a conex\303\243o entre essas ra\303\255zes e as ra\303\255zes caracter\303\255sticas, uma conex\303\243o que se torna clara somente na luz do operador de aniquila\303\247\303\243o. Portanto, obtemos a forma correta da solu\303\247\303\243o particular comparando as solu\303\247\303\265es homog\303\252neas das equa\303\247\303\265es diferenciais original e modificada. Essas solu\303\247\303\265es s\303\243o

QyQ+SSNZaEc2Ii1JJHJoc0clKnByb3RlY3RlZEc2Iy1JJ2Rzb2x2ZUdGJTYkLy1JJGxoc0dGKDYjSSNERUdGJSIiIS1JInlHRiU2I0kidEdGJSIiIg==PkkjWUhHNiItSSRyaHNHJSpwcm90ZWN0ZWRHNiMtSSdkc29sdmVHRiQ2JEkmbmV3ZGVHRiQtSSJ5R0YkNiNJInRHRiQ=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNZaEdGKCwmKihJJF9DMUdGKCIiIi1JJGV4cEdGJTYjLCRJInRHRighIiJGMS1JJHNpbkdGJTYjLCRGNiIiJEYxRjEqKEkkX0MyR0YoRjFGMkYxLUkkY29zR0YlRjpGMUYxNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNZSEdGKCwqKihJJF9DMUdGKCIiIi1JJGV4cEdGJTYjLCRJInRHRighIiJGMS1JJHNpbkdGJTYjLCRGNiIiJEYxRjEqKEkkX0MyR0YoRjFGMkYxLUkkY29zR0YlRjpGMUYxKipJJF9DM0dGKEYxRjJGMUY4RjFGNkYxRjEqKkkkX0M0R0YoRjFGMkYxRj9GMUY2RjFGMTcjRi4=

A hip\303\263tese correta para a forma da solu\303\247\303\243o particular \303\251 obtida pela extra\303\247\303\243o da solu\303\247\303\243o homog\303\252nea da equa\303\247\303\243o diferencial original da solu\303\247\303\243o geral da nova equa\303\247\303\243o. Essa separa\303\247\303\243o pode ser executada usando o comando FromLargeSolutionDeleteSmall, e resulta em

L0kjWXBHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQtST1Gcm9tTGFyZ2VTb2x1dGlvbkRlbGV0ZVNtYWxsR0YkNiRJI1lIR0YkSSNZaEdGJDckSSRleHBHRidJInRHRiQ=

L0kjWXBHNiIqKCwmKiYmSSJjR0YkNiMiIiJGKy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjLCRJInRHRiQiIiRGK0YrKiYmRik2IyIiI0YrLUkkc2luR0YuRjFGK0YrRitGM0YrLUkkZXhwR0YuNiMsJEYzISIiRis=

que \303\251 a solu\303\247\303\243o teste apropriada para o m\303\251todo dos coeficientes indeterminados

<Text-field style="Heading 1" layout="Heading 1">Coment\303\241rios Sobre as Novas Ferramentas</Text-field>

O pacote do Maple DEtools cont\303\251m os comandos diffop2de, e de2diffop. O primeiro converte um "operador diferencial" em uma equa\303\247\303\243o diferencial; o segundo, uma equa\303\247\303\243o diferencial em um "operador diferencial". O operador diferencial n\303\243o \303\251 mostrado como um operador, mas representado como um polin\303\264mio em algum s\303\255mbolo tal como LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjRHRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. Desse modo, dada a equa\303\247\303\243o diferencial

SSNERUc2Ig==

o operador diferencial encarnado pelo lado esquerdo \303\251 dado no Maple por

LUkqZGUyZGlmZm9wRzYiNiUtSSRsaHNHJSpwcm90ZWN0ZWRHNiNJI0RFR0YkLUkieUdGJDYjSSJ0R0YkNyRJI0R0R0YkRi4=

LCgqJEkjRHRHNiIiIiMiIiJGJEYmIiM1Ric=

Por outro lado, a equa\303\247\303\243o diferencial gerada pela aplica\303\247\303\243o do "operador diferencial"

PkkiUUc2IiwoKiRJJ2xhbWJkYUdGJCIiIyIiIkYnRigiIzVGKQ==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJRR0YoLCgqJEknbGFtYmRhR0YoIiIjIiIiRjBGMSIjNUYyNyNGLg==

para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== \303\251

LUkqZGlmZm9wMmRlRzYiNiVJIlFHRiQtSSJ5R0YkNiNJInRHRiQ3JEknbGFtYmRhR0YkRio=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMsKC1JJWRpZmZHRiY2JC1JInlHRig2I0kidEdGKC1JIiRHRiY2JEYzIiIjIiIiLUYuNiRGMEYzRjdGMCIjNTcjLCgtRi42JEY5RjNGOEY5RjdGMEY7

O operador que aniquilou o termo n\303\243o homog\303\252neo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== \303\251 gerado pela aplica\303\247\303\243o for\303\247ada brutamente do comando diffop2de. "Operadores" lineares de ordens bem maiores s\303\243o convertidos em equa\303\247\303\265es diferencias em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg==, e em cada est\303\241gio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg== \303\251 substitu\303\255do por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGNEY3Rj5GPkY+RitGPg==, a express\303\243o resultante sendo testada para desaparecimento id\303\252ntico como uma identidade em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. (Esse passo usa o comando do Maple solve/identity, e fornece os coeficientes no operador de aniquila\303\247\303\243o diferencial.)

O comando displayop, escrito para essa aplica\303\247\303\243o pelo Dr. Laurent Bernardin do Maplesoft, exibe o aniquilador como um operador diferencial no s\303\255mbolo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYsNiVRImRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiw2JVEjZHRGJ0Y1RjgvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkMvJSliZXZlbGxlZEdRJmZhbHNlRidGKy9GOVEnbm9ybWFsRic=. Esse comando assume que a vari\303\241vel independente \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, uma hip\303\263tese que \303\251 atualmente feito primeiro no comando AnnihilatingOperator. Dada a dificuldade em distinguirmos programaticamente entre os termos n\303\243o-homog\303\252neos 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 e 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, n\303\263s arbitrariamente optamos por fazer a vari\303\241vel independente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, sempre. (A alternativa poderia exigir a declara\303\247\303\243o expl\303\255cita da vari\303\241vel independente como parte da lista de argumento.)

O comando ApplyOperator \303\251 somente um complemento para o comando diffop2de, escondendo a diferen\303\247a entre a forma exibida de um operador e a forma "polinomial" exigida pelo Maple.

O comando ExtractCharacteristicRoots utiliza o comando de2diffop para gerar uma representa\303\247\303\243o polinomial da equa\303\247\303\243o diferencial homog\303\252nea. \303\211 claro, isso \303\251 somente a equa\303\247\303\243o caracter\303\255stica, ent\303\243o as ra\303\255zes caracter\303\255sticas s\303\243o imediatas.

O comando ExtractLatentRoots aplica o comando AnnihilatingOperator ao termo n\303\243o-homog\303\252neo, resultando em uma representa\303\247\303\243o polinomial do aniquilador. Os zeros desse polin\303\264mio s\303\243o ra\303\255zes latentes.

Finalmente, para extrair

SSNZaEc2Ig==

LCYqKEkkX0MxRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEkidEdGJSEiIkYmLUkkc2luR0YpNiMsJEYuIiIkRiZGJiooSSRfQzJHRiVGJkYnRiYtSSRjb3NHRilGMkYmRiY=

a solu\303\247\303\243o homog\303\252nea da ED original, de

SSNZSEc2Ig==

LCoqKEkkX0MxRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEkidEdGJSEiIkYmLUkkc2luR0YpNiMsJEYuIiIkRiZGJiooSSRfQzJHRiVGJkYnRiYtSSRjb3NHRilGMkYmRiYqKkkkX0MzR0YlRiZGJ0YmRjBGJkYuRiZGJioqSSRfQzRHRiVGJkYnRiZGN0YmRi5GJkYm

a solu\303\247\303\243o homog\303\252nea da ED modificada, devemos fazer mais do que simplesmente subtrair porque n\303\243o existe garantia que as mesmas constantes arbitr\303\241rias ir\303\243o se multiplicar como termos nas duas solu\303\247\303\265es. Para lidar com essa dificuldade, o comando FromLargeSolutionDeleteSmall estabelece todos os coeficientes para 1, converte cada solu\303\247\303\243o para um conjunto de fun\303\247\303\265es base, e executa uma diferen\303\247a de conjunto. Uma combina\303\247\303\243o linear das fun\303\247\303\265es resultanttes \303\251 ent\303\243o a solu\303\247\303\243o teste.

Uma palavra final para o leitor: Os seis comandos do Maple discutidos nesse artigo n\303\243o s\303\243o "\303\240-prova-de-balas". Qualquer um desses comandos pode ser facilmente quebrado ao aplic\303\241-los a argumentos inapropriados. Como at\303\251 mesmo uma inspe\303\247\303\243o superficial mostraria, nenhuma tentativa de checagem de digita\303\247\303\243o foi feita, ent\303\243o "lixo dentro, lixo fora" \303\251 definitivamente um aviso a ser ouvido. Deixamos ao leitor melhorar a nossa codifica\303\247\303\243o - nosso prop\303\263sito foi meramente demosntrarque o Maple j\303\241 possui as ferramentas rudimentares sobre as quais uma pedagogia superior poderia ser constru\303\255da.

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