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Math Help - How to solve d^2y/dx^2 = k^2y

  1. #1
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    How to solve d^2y/dx^2 = k^2y

    How do you solve

    d^2y/dx^2 = k^2 y

    for boundary conditions:

    y = y0 when x =0
    y = 0 when x = infinity

    Any help would be much appreciated.
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  2. #2
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    Re: How to solve d^2y/dx^2 = k^2y

    Quote Originally Posted by dbbwgr54 View Post
    How do you solve

    d^2y/dx^2 = k^2 y

    for boundary conditions:

    y = y0 when x =0
    y = 0 when x = infinity

    Any help would be much appreciated.
    Dear dbbwgr54,

    Use the trial solution y = Ae^{mx}.

    \Rightarrow\frac{dy}{dx}=Ame^{mx}

    \Rightarrow\frac{d^{2}y}{dx^2}=Am^{2}e^{mx}

    Therefore the auxiliary equation will be,

    m^2-k^2=0

    \Rightarrow m = \pm k

    Hope you can continue.
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  3. #3
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    Re: How to solve d^2y/dx^2 = k^2y

    Thank you Sudharaka, you help is much appreciated.

    I think I follow your procedure. Would A = y0?
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  4. #4
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    Re: How to solve d^2y/dx^2 = k^2y

    Quote Originally Posted by dbbwgr54 View Post
    Thank you Sudharaka, you help is much appreciated.

    I think I follow your procedure. Would A = y0?
    Not quite. You actually have to include both solutions, at first, and then use your boundary conditions to determine the two arbitrary constants. Remember: the solution to a second-order DE should always two arbitrary constants. So what will your general solution, with two arbitrary constants, look like?
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  5. #5
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    Re: How to solve d^2y/dx^2 = k^2y

    I think I'm lost. Sorry for this but I do not understand how the auxiliary equation gives m^2 - k^2.

    Also, is this treated as a homogeneous or in-homogeneous equation?


    Thanks for the help!
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  6. #6
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    Re: How to solve d^2y/dx^2 = k^2y

    Quote Originally Posted by dbbwgr54 View Post
    How do you solve

    d^2y/dx^2 = k^2 y

    for boundary conditions:

    y = y0 when x =0
    y = 0 when x = infinity

    Any help would be much appreciated.
    The DE can be written as \frac{d^2 y}{dx^2} - k^2 y = 0. It is obviously homogeneous.

    Now refer to your class notes and textbook on how to solve second order DE's with constant coefficients.

    Alternatively, you can substitute \frac{d^2 y}{dx^2} = \frac{d \left( \frac{v^2}{2}\right)}{dy} where v = \frac{dy}{dx} and then integrate directly with respect to y and then solve the resulting 1st order DE.

    As always, the techniques, applications and examples are found in the class notes and textbook.
    Last edited by Ackbeet; November 8th 2011 at 01:13 PM. Reason: typo
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