Homogeneous second-order ODE with non-constant coefficients

Question

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 :)

Answer 1

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

$$ y = \frac{C_1}{1+x^2}+\frac{C_2x}{1+x^2} $$

Answer 2

Let $u=x^2+1$. Then $u'=2x$ and $u''=2$. Clearly then $(1+x^2)y'' + 4xy' + 2y = 0$ can be rewritten $uy'' + 2u'y' + u''y=0$. This is equivalent to $(uy)''=0$. Now just integrate twice. $(uy)' = C$ and $uy = Cx + D$. We know $u=x^2+1$ so $(x^2+1)y = Cx+D$ or if you prefer $y=\frac{Cx + D}{x^2+1}$.