Solution of the differential equation $ddotx(t)+sin(omega t)x(t)=cos[eta(t)]$

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The differential equation:



$$ddotx(t)+sin(omega t)x(t)=cos(eta t)$$
has an analytical solution involving Mathieu functions.
This is valid if both $omega$ and $eta$ are constant.
Suppose the term $eta$ is a normal distribuited random variable $eta(t)$ with mean $mu$ equal to zero and variance $sigma^2$. How changes the solution?
Or better: Is it possible to find analytically the spectrum of $x(t)$? Thanks










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    up vote
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    favorite
    1












    The differential equation:



    $$ddotx(t)+sin(omega t)x(t)=cos(eta t)$$
    has an analytical solution involving Mathieu functions.
    This is valid if both $omega$ and $eta$ are constant.
    Suppose the term $eta$ is a normal distribuited random variable $eta(t)$ with mean $mu$ equal to zero and variance $sigma^2$. How changes the solution?
    Or better: Is it possible to find analytically the spectrum of $x(t)$? Thanks










    share|cite|improve this question























      up vote
      1
      down vote

      favorite
      1









      up vote
      1
      down vote

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      1






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      The differential equation:



      $$ddotx(t)+sin(omega t)x(t)=cos(eta t)$$
      has an analytical solution involving Mathieu functions.
      This is valid if both $omega$ and $eta$ are constant.
      Suppose the term $eta$ is a normal distribuited random variable $eta(t)$ with mean $mu$ equal to zero and variance $sigma^2$. How changes the solution?
      Or better: Is it possible to find analytically the spectrum of $x(t)$? Thanks










      share|cite|improve this question













      The differential equation:



      $$ddotx(t)+sin(omega t)x(t)=cos(eta t)$$
      has an analytical solution involving Mathieu functions.
      This is valid if both $omega$ and $eta$ are constant.
      Suppose the term $eta$ is a normal distribuited random variable $eta(t)$ with mean $mu$ equal to zero and variance $sigma^2$. How changes the solution?
      Or better: Is it possible to find analytically the spectrum of $x(t)$? Thanks







      differential-equations stochastic-processes special-functions stochastic-calculus






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      asked Sep 5 at 8:39









      Riccardo.Alestra

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