Nos Problemas 9 a 12, encontre a solução do problema de valor inicial dado.
y = sen(x^2)

Passo 1

Beleza, vamos lá! Temos a equação diferencial com a condição inicial . Isso aí faz parte das equações diferenciais lineares de primeira ordem. Para resolver isso, a gente vai precisar usar o método do fator integrante.

Passo 2

Primeiro, vamos identificar a equação na forma padrão: , onde e . Isso ajuda a gente a aplicar o método certo.

Passo 3

Agora, vamos encontrar o fator integrante. Ele é dado por . Como , temos:

Passo 4

A gente multiplica toda a equação diferencial pelo fator integrante:
Isso simplifica a equação para uma forma que pode ser integrada facilmente.

Passo 5

Perceba que o lado esquerdo agora é a derivada do produto . Então podemos escrever:
Agora, vamos integrar ambos os lados em relação a :

Passo 6

Isolando , temos:
Agora, vamos usar a condição inicial para encontrar a constante :
Como , então:

Resposta

A solução da equação diferencial é:

Exercícios relacionados

Se , mostre que
<html><head></head><body><div>Seja</div>
<div><span class=“math-block “>
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<div>Determine:<br>
a) Domínio</div>
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Verifique se são convergentes:
<html><head></head><body><div>Escreva os primeiros cinco termos das sequências abaixo. Determine se estas convergem ou divergem. Se convergirem, determine o limite.<br>
<span class="math-inline ">
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