Consider the second order differential equation $$y''-x^2y=0 $$ where $y$ itself is a function of $x$. I do not know how to solve this equation. I tried a series expansion and failed, and because the coefficients are not constant, I can not use the characteristic equation to solve it either. Hence, here I am, looking for any hints on how to solve this equation for $y$.
I know there are tons of questions already out there concerning second order differential equations looking like this one, and I looked through just about every one of them, however all the solutions provided seem to be very situational for the given DE, and I have yet to find a general method that I can use to solve the above. I though about reducing the order of the equation.