I'm trying to solve this ODE using power series: $$(y+x^2)y''+y=0$$ I tried to solve it, keeping in mind that $x=0$ is a singular point, then substituting $y=\sum_{n=0}^\infty a_nx^n$, but I didn't get anything helpful. I've also checked that it isn't equidimensional or scale-invariant.
I have been asked to express the solution as a linear combination of two power series, then compute the radius of convergence. Do I need to solve the ODE anyway?
Let $u=y+x^2$ ,
Then $u'=y'+2x$
$u''=y''+2$
$\therefore u(u''-2)+u-x^2=0$
$uu''-u-x^2=0$
$u''=\dfrac{x^2}{u}+1$
You can consider as two members Emden-Fowler type nonlinear ODE and follow the method in http://www.sciencepubco.com/index.php/ijamr/article/download/723/628