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Math Help - Second order differential equation with non constant coefficients.

  1. #1
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    Second order differential equation with non constant coefficients.

    Hi all,

    While solving some physics problem I came across a differential equation.

    \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+c_1x^2y=c_2y

    Where all the constants c_1 and c_2 are positive. I need a solution for this equation. Please indicate me how to go about this. It will be helpful if the solution is in closed form. It might be helpful to know that, the solution to,

    \frac{\mathrm{d}^2y}{\mathrm{d}x^2}-c_1x^2y=c_2y

    is a product of hermite polynomials with a Gaussian envelope.

    Thanking you.
    Shantanu.
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  2. #2
    MHF Contributor

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    Re: Second order differential equation with non constant coefficients.

    In general, solutions to even linear differential equations, with variable coefficients, cannot be written in terms of "elementary" functions. If you already know that solutions to the second equation are "products" of Hermitian functions, why not try writing the solutions to the first equation that way with argument ix rather than x? Failing that, some sort of series solution would be called for.
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