# Problem with a change of variable in ode

I have this differential equation

$$x(1-x^2)y''-(1-x^2)^2y'+5x^3y=0,$$

for $-1<x<1$. The hint claims that we should use the change of coordinates $t=-\frac{1}{2}\ln(1-x^2)$, and this will transform the ODE into a constant coefficient one. However, after the sustitution:

$$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{x}{1-x^2}\frac{dy}{dt}$$

and

$$\frac{d^2y}{dx^2}=\frac{d^2y}{dt^2}\left(\frac{dt}{dx}\right)^2+\frac{dy}{dt}\frac{d^2t}{dx^2}=\frac{x^2}{(1-x^2)^2}\frac{d^2y}{dt^2}+\frac{1+x^2}{(1-x^2)^2}\frac{dy}{dt},$$

and replacing into the original ODE, I get NOT a constant coefficient equation as desired.

I don't know what detail I have missed or if I took a wrong approach. Any suggestions?

Thanks

EDIT: Working backwards, from the constant-coefficient differential equation $a \dfrac{d^2 y}{dt^2} + b \dfrac{dy}{dt} + c y = 0$ the suggested change of variables gives you $$a x (1 - x^2)^2 y'' - (1-x^2)(a + (a-b) x^2) y' + c x^3 y = 0$$ so perhaps the intended differential equation was $$x (1-x^2)^2 y'' - (1-x^2)^2 y' + 5 x^2 y = 0$$