Discuss about constant coefficient odes

Question

Using substitution to simplify an ODE

Solve the following problem:

Let a, B be constants. Consider the ODE

y'' + (a/x)y' + (B/x^2)y = 0

on the half-line x belongs to (0,infinity).

(a) Make the substitution t = ln(x) and write the ODE with independent variable t.

(b) From what you know about constant coefficient ODEs, separate the problem into three cases according to the values of a and B.

(c) In each of the three cases, give the general solution to the original ODE (that is,in terms of x, not t).

Show all work.