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    Using substitution to simplify an ODE

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    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).

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    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.