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).
One common technique for solving ODEs is to make a substitution which transforms the problem into a new, simpler ODE in a new variable. This simpler ODE is then solved, and finally back substitution is used to obtain the solution to the original problem. This solution comprises a single page, written in Word with equations in Mathtype, illustrating the application of this technique for the given question.