Find the general solution of the differential equation
y′ =$\displaystyle \frac{2x^2+y^2}{xy}$. in explicit form.
Use a substitution of $\displaystyle u = \frac{y}{x} $
$\displaystyle y = xu $
$\displaystyle \frac{dy}{dx} = u + x\frac{du}{dx} $
$\displaystyle \frac{dy}{dx} = \frac{2x^2}{xy}+\frac{y^2}{xy}$
$\displaystyle \frac{dy}{dx} = \frac{2x}{y}+\frac{y}{x}$
$\displaystyle u + x\frac{du}{dx} = \frac{2}{u}+u$
$\displaystyle x\frac{du}{dx} = \frac{2}{u}$
$\displaystyle \frac{du}{u} = 2 \times \frac{dx}{x}$