Find a simple solution by guessing and find a recursion

Question

Consider the differential equation

(x² + 1)y" + xy' - y = 0.

(a) Find a simple solution by guessing.

(b) Now look for a solution in form of a power series solution, i.e. substitute ∑^∞_n=0 a_nxⁿ into the equation.

(c) Find a recursion relation for the coefficients of the power series solution.

(d) Show that the recursion relation yields the simple solution of (a).

(e) Show that the recursion relation also yields a second solution and give its first three non-zero terms.

(f) Check that the two solutions are linearly independent.