Explicit solution of a second order differential equation

Question

Let $y:[0,1]\mapsto \mathbb{R}$. I'd like to know the existence of the explicit solution of the differential equation $$ y^{\prime\prime}-xy^\prime-x^2y=3x-x^3 $$ with the boundary conditions $y(0)=y(1)=0$. I had tried to substitute $x=e^t$ but I could not change to the linear form. Could anyone give me a suggestion?

Answer 1

It does not look good. Maple produces a mess involving integrals of rational expression involving Kummer's hypergeometric functions. Totally useless.

Numerical solution, also found with Maple, looks this way:

Answer 2

Let $y=u+x$ ,

Then $y'=u'+1$

$y''=u''$

$\therefore u''-x(u'+1)-x^2(u+x)=3x-x^3$ with $u(0)=0$ , $u(1)=-1$

$u''-xu'-x-x^2u-x^3=3x-x^3$ with $u(0)=0$ , $u(1)=-1$

$u''-xu'-x^2u=4x$ with $u(0)=0$ , $u(1)=-1$