Consider the following linear differential equation.
$$\frac{dw^2}{dz^2}+(\frac{1}{3z}+\frac{1}{6(z+1)})\frac{dw}{dz}+(-\frac{1}{3z^2}-\frac{1}{6(z+1)^2}+\frac{1}{2z(z+1)})w=0$$ Let $t=1/z$ and write the equation in $w$ and $t$ and solve the equation.
I am stuck solving the equation.
I can see that $\frac{dw}{dz}=\frac{dw}{dt}(-t^{-2})$, $\frac{d^2w}{dz^2}=t^{-4}\frac{d^2w}{dt^2}$ and $\frac1{z+1}=\frac{t}{1+t}$.
I guess the new equation is also Fuschian so we need to find the singularities and the indicial equations, but I can't see how.