Question

Resolva o problema de valor inicial de acordo com dos dados fornecidos e assinale a alternativa que contém a resposta CORETA

{ y`-2xy = 2x
y(0) = 3

Answer 1

Olá

$\displaystyle \mathsf{y'-2xy=2x}\\\mathsf{y(0)=3}\\\\\\\mathsf{ \frac{dy}{dx}-2xy=2x }\\\\\\\mathsf{ \frac{dy}{dx}=2x+2xy }\\\\\\\mathsf{ \frac{dy}{dx}=2x(1+y) }\\\\\\\mathsf{ \frac{dy}{1+y}=2xdx }\\\\\\\text{Integrando dos dois lados}\\\\\\\mathsf{ \int \frac{dy}{1+y}=\int2xdx }\\\\\\\mathsf{\ell n|1+y|=x^2+C}\\\\\\\text{condicao do valor inicial y=3, x=0}\\\\\mathsf{\ell n|1+3|=0^2+C}\\\\\\\mathsf{C=\ell n |4|}$

$\displaystyle \mathsf{\ell n|1+y|=x^2+\ell n |4|}\\\\\text{Aplicando propriedade de ln para isolarmos o y}\\\\\\\mathsf{\ell n|1+y|=\ell n |e^{x^2+\ell n|4|}|}\\\\\\\mathsf{\ell n|1+y|=\ell n |4e^{x^2}|}\\\\\\\text{bases iguais, podemos cancelar o log}\\\\\\\mathsf{1+y=4e^{x^2}}\\\\\\\boxed{\mathsf{y=4e^{x^2}}-1} \qquad \qquad \Longleftarrow \quad \text{Resposta}$

Answer 2

Resposta correta: y = 4ex² - 1