Question

Dw/dx = x*w²*sen(x²) com w(0) = 1

Answer 1

$\displaystyle \frac{dw}{w^2}=x\sin x^2 dx\\ \\ \int\frac{dw}{w^2}=\int x\sin x^2 dx\\ \\ -\frac{1}{w}=-\frac{1}{2}\cos x^2+C\\ \\ -\frac{1}{w(0)}=-\frac{1}{2}\cos 0+C\\ \\ -1=-\frac{1}{2}+C\\ \\ C=-\frac{1}{2} \\\\ \boxed{-\frac{1}{w}=-\frac{1}{2}\cos x^2-\frac{1}{2}}$

Answer 2

dw/dx = x w² sen(x²) ⇒ w(x) = ∫ x w² sen(x²) dx = w² ∫ xsen(x²) dx
Resolvendo a integral ⇒ caso queira mais detalhes eu resolvo passo a passo
Temos, w² -1/2 cos (x²) = -1/2 w² cos(x²) ⇒ w(x) = -1/2 w² cos (x²)

w(0)=1 ⇒ 1 = -1/2 w²  cos(0) ⇒ 2 = -1 w² 1 ⇒ w² = -2 

dw/dx = x -2 sen x² = -2x sen(x²)