I have to solve this DE
((x+2)^2)y'' -2y=4
the hint says to get rid of the x+2 but I have no idea where to start from... do I have to replace x+2 with another variable or what?
I would replace the $\displaystyle x+2$ with $\displaystyle t=x+2,$ and you get
$\displaystyle \dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}=\dfrac {dy}{dx}.$ Hence, the DE becomes
$\displaystyle t^{2}\,\dfrac{d^{2}y}{dt^{2}}-2y=4.$
This equation is a Cauchy-Euler equation.