I have this differential equation : $$4x^{2}y''+8x^{3}y'+(4x^{2}-3)y = 0$$ and it is given that the equation has a solution $$y_{1}(x)=x^{r}.$$
I set $y_2$ as $$ y_{2}=x^{r}v(x).$$
When I'm trying to find the first and second derivative I start to fail.
I get $$ y'_2=x^{r-1}(xv'+rv)$$ but the second is always long and makes no sense when I keep on with the problem. After I have put derivatives for the $y$, $y'$ and $y''$ in the original equation I get complex equations where i see no way to use reduction of order.
Can someone please help me and tell me what I'm doing wrong.