 Solution

For $w^{\prime \prime }+P(z)w^{\prime }+Q(z)w=0$, a finite point $z_0$ is regular singular whenextend analytically to $z_0$. Inspection shows that $a,b,c$ satisfy these conditions. There are no other finite singularities; with the Fuchs relation on the six exponents, the point at infinity is ordinary in the corresponding sphere description.