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Math Help - solve the following differential equation

  1. #1
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    solve the following differential equation

    I am supposed to solve the following

    zy'' + (z-2)y' -3y = 0 and y=z^3 is a solution.

    I have a feeling I should use the Wronskian but I don't know how because my text book is horrible for explaining anything.
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  2. #2
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    Quote Originally Posted by sarah232 View Post
    I am supposed to solve the following

    zy'' + (z-2)y' -3y = 0 and y=z^3 is a solution.

    I have a feeling I should use the Wronskian but I don't know how because my text book is horrible for explaining anything.
    You can but an easy way is let y = z^3 u.

    So y' = z^3 u' + 3z^2 u and y'' = z^3u'' + 6z^2u' + 6z u. Now substitute into your ODE and it becomes

    z(z^3u'' + 6z^2u' + 6z u)+(z-2)(z^3 u' + 3z^2 u) - 3 z^3 u = 0

    which becomes, after some simplification

    z u'' + (z+4)u' = 0

    If you let v = u' then we obatin z v' + (z+4)v = 0 (separable). Integrate, put back in u' and integrate again to get u. Then y = z^3 u.
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  3. #3
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    Do you mind showing me the Wronskian method? I believe I will need to know this for my exam. Thanks!
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  4. #4
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    Quote Originally Posted by sarah232 View Post
    Do you mind showing me the Wronskian method? I believe I will need to know this for my exam. Thanks!
    Here a post from this forum on that http://www.mathhelpforum.com/math-he...e-problem.html
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