Resolver as seguintes EDO 4 x y^2 dx + (x^2 + 1) dy = 0 PVI: y(0) = 1

Passo 1

Beleza, vamos resolver essa Equação Diferencial Ordinária (EDO) que está nessa receita aqui: com a condição inicial . O conteúdo que vamos usar envolve equações diferenciais e, mais especificamente, a técnica de identificar se a equação é exata ou se precisamos encontrar um fator integrante para torná-la exata. Vamos precisar entender conceitos básicos de derivadas parciais e integrais para resolver isso.

Passo 2

Primeiro, vamos verificar se a equação é exata. Uma equação diferencial na forma é exata se . Aqui, e .

Passo 3

Calculando as derivadas parciais:
Como , a equação não é exata.

Passo 4

Quando a equação não é exata, podemos procurar um fator integrante que depende apenas de ou de . Vamos tentar um fator que depende apenas de , ou seja, . Para isso, usamos a relação:
Se isso for uma função de apenas, então podemos integrá-la para encontrar .

Passo 5

Simplificando a expressão:
Vemos que ainda tem na expressão, o que indica que um fator que depende apenas de não funcionará. Vamos tentar um fator que depende de , ou seja, .

Passo 6

Usamos a relação para :
Como isso depende apenas de , podemos integrar para encontrar .

Passo 7

Integrando:
Portanto, o fator integrante é .

Passo 8

Multiplicando a equação original pelo fator integrante:
Simplificando:
Agora, a equação é exata.

Passo 9

Agora, sabemos que existe uma função tal que:
Integrando em relação a :
Onde é uma função que depende apenas de .

Passo 10

Derivando em relação a e igualando a com :
Simplificando:
Isso implica que:
Integrando :
Essa integral pode ser resolvida por substituição , mas para simplificar, vamos considerar que já foi integrada.

Passo 11

Assim, a solução geral é:
Onde é a constante de integração.

Passo 12

Aplicando a condição inicial :
Simplificando:
Então, .

Resposta

A solução da EDO com a condição inicial é:

Exercícios relacionados

Resolver as seguintes EDO (2+y) dx - (3-x) dy = 0 Solução Geral
Resolver as EDO d y/d x - y/x = x
<html><head></head><body><div>Resolver as seguintes EDO<span class="math-inline ">
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</body></html>
<html><head></head><body><div>Resolver as seguintes EDO<span class="math-inline ">
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97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z”></path></g></g></g><g data-mml-node=“mo” transform=“translate(1008.6,0)”><path data-c=“2B” d=“M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z”></path></g><g data-mml-node=“mn” transform=“translate(1786.6,0)”><path data-c=“34” d=“M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z”></path></g></g><rect width=“1816.8” height=“60” x=“120” y=“220”></rect></g></g></g></svg></mjx-container></span> Solução Geral</div>
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