Solu\303\247\303\265es gr\303\241ficas de equa\303\247\303\265es diferenciais 1) Plotando v\303\241rias solu\303\247\303\265es de equa\303\247\303\265es diferenciais com seq e plot Uma das grandes vantagens do uso do Maple \303\251 a facilidade de se plotar gr\303\241ficos de solu\303\247\303\265es de equa\303\247\303\265es diferenciais. Podemos plotar uma ou mais solu\303\247\303\265es de uma equa\303\247\303\243o diferencial de v\303\241rios modos dentro do Maple V. Descrevemos a seguir tr\303\252s modos de se plotar gr\303\241ficos de equa\303\247\303\265es diferenciais O Maple n\303\243o resolve simultaneamente uma equa\303\247\303\243o diferencial com mais de uma condi\303\247\303\243o inicial. Um modo de plotar uma fam\303\255lia de solu\303\247\303\265es de uma equa\303\247\303\243o diferencial, com mais de uma condi\303\247\303\243o inicial, \303\251 gerar uma sequ\303\252ncia de condi\303\247\303\265es iniciais e depois gerar uma lista de solu\303\247\303\265es com o comando dsolve. Vmaos usar o comando seq para criar uma sequ\303\252ncia de condi\303\247\303\265es iniciais seq( f(i), i = m..n) na qual \303\251 uma fun\303\247\303\243o e i varia de m at\303\251 n, m e n inteiros. 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bmhvfmRhc35saW5oYXN+dGFuZ2VudGVzLkYnRi9GTkY5Rk9GPkZARkJGREZQRlJGYXFGLw== 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Com o comando phaseportrait da biblioteca DEtools o Maple plota o Retrato de Fase da equa\303\247\303\243o nos pontos e dom\303\255nios especificados. Este comando plota automaticamente tamb\303\251m o Campo de Dire\303\247\303\265es das solu\303\247\303\265es . Chamamos aten\303\247\303\243o de que o Maple emprega m\303\251todos num\303\251ricos para plotar as solu\303\247\303\265es e, portanto, n\303\243o \303\251 permitido par\303\242metros literais na equa\303\247\303\243o dada . A sintaxe do comando \303\251 dada por phaseprotrait( D(y)(x)=f(x,y), [y], x=a..b,{ [x1,y1], ..., [xN,yN] }, y=c..d , OutrasCaracter\303\255sticas=op\303\247\303\243o) Vamos plotar as solu\303\247\303\265es da equa\303\247\303\243o y' = -2y + exp(x) nos 6 pontos especificados abaixo 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 Dentre as outras op\303\247\303\265es que est\303\243o descritas em DEtools[DEplot] temos iterations, arrows, dirgrid, obsrange, scene, colour, linecolour, stepsized Podemos usar ainda as op\303\247\303\265es dsolve/numeric (por exemplo, method = ) e de ?plot[options] (por exemplo, title = `equa\303\247\303\243o1` ). Damos um exemplo no qual invertemos os eixos com scene = [y(x),x] 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2I1EhRictRiw2JVEmc2NlbmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tSShtZmVuY2VkR0YkNiYtRiM2KC1GLDYlUSJ5RidGMkY1LUZTNiQtRiM2JS1GLDYlUSJ4RidGMkY1LyUrZXhlY3V0YWJsZUdGQEY8RjwtRjk2LVEiLEYnRjxGPi9GQkY0RkNGRUZHRklGSy9GTlEmMC4wZW1GJy9GUVEsMC4zMzMzMzMzZW1GJ0ZobkZbb0Y8RjwvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGXW8tRjk2LVEifkYnRjxGPkZBRkNGRUZHRklGS0Zhby9GUUZiby1GLDYlUSdtZXRob2RGJ0YyRjVGOC1GLDYlUSpjbGFzc2ljYWxGJ0YyRjUtRlM2Ji1GIzYlLUYsNiVRKWZvcmV1bGVyRidGMkY1RltvRjxGPEZlb0Zob0Zdb0ZbcC1GLDYlUSlzdGVwc2l6ZUYnRjJGNUY4LUkjbW5HRiQ2JFEkLjA1RidGPEZdb0Zbb0Y8 linecolour=sin(t*Pi/2), title=`Solucao Invertida`, 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 1) ?plot[options] que s\303\243o axesfont, labelfont, linestyle, thickness, titlefont, resolution, xtickmarks, axes, color, colour, coords, font, scaling, style, symbol, view ,` 2) dsolve/numeric que s\303\243o procedure, value, corrections, errorper,s tartinit, stepsize, tolerance, abserr, control, ctrl, itask, maxder, maxfun, maxkop, maxord, maxpts, maxstep, method, minrel, minstep, number, relerr, start, initial Observa\303\247\303\243o: Ao resolver uma equa\303\247\303\243o diferencial numericamente no Maple V, num intervalo arbitrariamente escolhido, podemos estar escolhendo um intervalo maior do que o dom\303\255nio te\303\263rico da solu\303\247\303\243o procurada. Neste caso o Maple V pode apresentar uma solu\303\247\303\243o que fica " maluca" ap\303\263s passar por um desses pontos limites do dom\303\255nio regular da fun\303\247\303\243o. Por exemplo, a equa\303\247\303\243o y' = - x / y tem como solu\303\247\303\243o geral . Se damos a condi\303\247\303\243o inicial , temos a solu\303\247\303\243o que tem o dom\303\255nio . Nos pontos e temos que . Nestes pontos a equa\303\247\303\243o diferencial n\303\243o est\303\241 definida pois a tangente \303\240 curva solu\303\247\303\243o se torna vertical. Vejamos o que ocorre quando tentamos calcular a solu\303\247\303\243o num dom\303\255nio com phaseportrait. O gr\303\241fico que deveria ser um arco de c\303\255rculo aparece prolongado Se restringimos o dom\303\255nio obtemos o arco de c\303\255rculo esperado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKERFdG9vbHNGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjpGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVkY9RkA= 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 Devemos ter cautela ao investigar solu\303\247\303\265es de equa\303\247\303\265es diferenciais numericamente com phaseportrait , pois, o que o Maple apresenta pode n\303\243o exprimir uma verdade matem\303\241tica. Isto pode trazer problemas e \303\251 muito comum quando a solu\303\247\303\243o geral s\303\263 pode ser expressa implicitamente. Um modo de resolver este problema \303\251 plotar as solu\303\247\303\265es implicitamente, conforme descrito a seguir. c) Plotando solu\303\247\303\265es impl\303\255citas - plots[contourplot , implicitplot] Vamos supor que queremos plotar as solu\303\247\303\265es da equ\303\247\303\243o diferencial que tem como solu\303\247\303\243o geral . Podemos tra\303\247ar as curvas de n\303\255vel das solu\303\247\303\265es no Maple com os comandos contourplot ou com implicitplot da bilioteca plots . Com o comando contourplot tra\303\247amos curvas de n\303\255vel de uma fun\303\247\303\243o dentro dos intervalos especificados. Com este comando n\303\243o podemos escolher quais curvas ser\303\243o tra\303\247adas, mas podemos especificar a precis\303\243o desejada com a op\303\247\303\243o grid=[m,n] (default = [25,25]) . No exemplo abaixo vamos preencher os espa\303\247os entre as curvas com filled=true "num intervalo de cores" que vai do azul ao vermelho com coloring = [red, blue] pois cada cor \303\251 dada por um n\303\272mero LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA 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 Outro modo de obter as curvas de n\303\255vel da fun\303\247\303\243o acima \303\251 com implicitplot . Para usar este comando temos que explicitar todas as curvas de n\303\255vel que queremos plotar. Um modo de obter a lista das curvas \303\251 usar o comando seq e depois usar implicitplot para plot\303\241-las 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA 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 Observa\303\247\303\243o: Um modo de se obter os valores das constates CJ = F(x_J,y_J) para cada condi\303\247\303\243o inicial \303\251 substituir (x_J,y_J) na solu\303\247\303\243o geral F(x,y) = C com subs e depois calcular CJ numericamente com solve e evalf. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv JSFH