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It's well known that the solution of the differential equation: $\ddot x(t)+\omega^2x(t)=\sin(\psi t)$ has the form: $x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi t)}{\psi^2-\omega^2}$ Obviously, if $\psi=\omega$, there is a resonance and the amplitude of the oscillations diverges.

My question is: what happens if $\psi$ is a normal distribuited random variable with mean value $\omega$ and variance $\sigma$? Thanks in advance.

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    A more interesting question would be to ask what sort of solution would be expected for a harmonic oscillator with a stochastic forcing term, for example,$\ddot x(t)+\omega^2x(t)=\sin(\omega t+\sigma W_t),$where $(W_t:t\geqslant 0)$ is a standard Wiener process.2013-10-06

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