(a) Derive a second linearly independent solution of (1) by reduction of order; but instead of using (9), Sec. 2.1, perform all steps directly for the present ODE (1).

(b) Obtain x^m ln x by considering the solutions x^m and x^m+s of a suitable Euler–Cauchy equation and letting s → 0.

(c) Verify by substitution that x^m ln x, m = (1 - α)/2, is a solution in the critical case.

(d) Transform the Euler–Cauchy equation (1) into an ODE with constant coefficients by setting x = e^t (x > 0).

(e) Obtain a second linearly independent solution of the Euler–Cauchy equation in the “critical case” from that of a constant-coefficient ODE.