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Math Help - 2nd order with variable coefficients

  1. #1
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    2nd order with variable coefficients

    Hello,

    I want to solve the following differential equation:

    y''(x) + (a+b*exp(-c*x)) y'(x) + (d+f*exp(-g*x)) y(x) = s(x)

    with y(0)=0, y'(0)=0
    a,b,c,d,f,g are constant and s(x) is known. Any suggestions on how I could solve it? I am even not able to solve the homogeneous equation (s(x)=0).

    To find a solution to the following approximation would also help me:
    y''(x) + (a-c*x) y'(x) + (d-g*x) y(x) = s(x)

    Looking forward to your answers.

    Regards,
    Heiko
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  2. #2
    MHF Contributor chisigma's Avatar
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    You can try to use a specific property of the Laplace Tranform that extablishes that...

    \displaystyle \varphi(s)= \mathcal{L} \{y(t)\} \implies \mathcal{L} \{e^{\alpha\ t}\ y(t)\} = \varphi(s-\alpha) \implies

    \displaystyle \implies \mathcal{L} \{e^{\alpha\ t}\ y^{'}(t)\} = (s-\alpha)\ \varphi(s-\alpha) - y(0) (1)

    ... and, considering that is y(0)= y^{'} (0)=0 the DE in terms of Laplace Transform becomes...

    \displaystyle (s^{2} + a\ s + d)\ \varphi(s) + b\ (s+c)\ \varphi(s+c) + f\ (s+g)\ \varphi(s+g) = \sigma(s) (2)

    ... where \sigma(s)= \mathcal{L} \{s(t)\}. The solution of (2) [a 'functional equation'...] is however not trivial ...

    Kind regards

    \chi \sigma
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