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Math Help - differential equation proving

  1. #1
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    differential equation proving

    Given a differential equation
    y"= -(2g'/g)y' + [V-E-(g"/g)]y

    V(x)= x^(2k) & g(x) = exp(-βx^2/2)

    subtitute V(x) & g(x), found

    y" = 2βxy' + [x^(2k) - (βx)^2 + β - E]y

    the problem i encounter is the subtitution I done is missing β

    y" = 2βxy' + [x^(2k) - (βx)^2 - E]y

    P/s: ^ represent power of
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  2. #2
    Super Member Aryth's Avatar
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    Well, it's actually just an error in differentiating:

    g(x) = e^{-\frac{\beta x^2}{2}}

    g'(x) = -\beta xe^{-\frac{\beta x^2}{2}}

    The first one was an exponential, so we used the chain rule, but notice that this is a product, so we must use the product rule:

    g''(x) = -\beta e^{-\frac{\beta x^2}{2}} + (\beta x)^2e^{-\frac{\beta x^2}{2}}

    Now we do the division:

    -\frac{g''(x)}{g(x)} = \frac{\beta e^{-\frac{\beta x^2}{2}} - (\beta x)^2e^{-\frac{\beta x^2}{2}}}{e^{-\frac{\beta x^2}{2}}}

    Which gives:

    -\frac{g''(x)}{g(x)} = \beta - (\beta x)^2

     = -(\beta x)^2 + \beta

    And there is the right answer.
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  3. #3
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    Thanks ...

    Thanks for helping me out. Guess I was kinda careless as well. Anyway, thanks again ...
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