ODE5.3

$xy''-y'=3x^2\,$

Make this substitution to reduce the ODE to first order:

Let $p = y'\,$

$xp'-p=3x^2\,$

$p'-\frac{1}{x}p = 3x\,$

Let the integrating factor $\rho = e^{-\frac{1}{x}\,dx} = e^{-\ln x} = \frac{1}{x}\,$

Multiply through by $\rho\,$ .

$\frac{1}{x} p' - \frac{1}{x^2} p = 3\,$

This is equivalent to

$\frac{d}{dx}\left[\frac{1}{x}p\right] = 3\,$

$\frac{1}{x}p = 3x + c_1\,$

$p = 3x^2 + c_1 x\,$

Since $p = y'\,$ ,

$y' = 3x^2 + c_1 x\,$

$y = x^3 + \frac{c_1}{2} x^2 + c_2\,$