x^3y' + 2y = x^3 + 2x y(1) = e+1
Which method would you use to solve this?
The equation is linear so first put it in standard form
$\displaystyle y' + \frac{2y}{x^3} = 1 + \frac{2}{x^2}$
The integrating factor is
$\displaystyle \mu = e^{-\frac{1}{x^2}} $ so
$\displaystyle \frac{d}{dx} e^{-\frac{1}{x^2}} \cdot y = \left( 1 + \frac{2}{x^2}\right) e^{-\frac{1}{x^2}} $ which integrates giving
$\displaystyle e^{-\frac{1}{x^2}} \cdot y = x e^{-\frac{1}{x^2}} + c $
I think you can take it from here.