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Math Help - Hermites differential equation

  1. #1
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    Hermites differential equation

    Hello all,

    I want to show that a function w is a solution to Hermite's equation

    \frac{d^{2}w}{dx^{2}}-2x\frac{dw}{dx}+2\ell w =0

    if and only if w is a solution to the Sturm-Liouville equation

    \left( e^{-x^{2}}w' \right)'+2\ell e^{-x^{2}}w=0

    Could someone assist me in the right direction.

    Thanks.
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  2. #2
    Member Black's Avatar
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    \left(e^{-x^2}w'\right)'+2\ell e^{-x^2}w=e^{-x^2}w''-2xe^{-x^2}w'+2\ell e^{-x^2}w=0.

    Divide both sides of the equation by the exponential and you get the first equation.
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