# Using substitution to simplify an ODE

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).

https://brainmass.com/math/ordinary-differential-equations/substitution-simplify-ode-170922

#### Solution Summary

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.