I have this non-constant coefficient homogeneous second ODE:
$$(1 + x^{2})y'' + 4xy' + 2y = 0.$$
I have found a power series solution for this equation but I am then asked to transform it to system of first order ODE and also fundamental matrix form for that. But i don’t know how to transform it. I can do it if that equation is originally with constant coefficient, but that is polynomial coefficient and i just don’t know how. I am also not sure if that equation is in Euler form or not since it has $1 + x^2$ instead of $x^{2}$
Any help would be appreciated. Thank you very much 🙂
Best Answer
Making the substitution
$$ y = \frac{x^\lambda}{1+x^2} $$
we get
$$ \lambda(\lambda-1)x^{\lambda-2} = 0 \Rightarrow \left\{ \begin{array}{rcl} \lambda & = & 0\\ \lambda & = & 1 \end{array}\right. $$
hence
$$ y = \frac{C_1}{1+x^2}+\frac{C_2x}{1+x^2} $$