# Harmonic oscillator with stochastic forcing

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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Then, for every $t\gt0$, $x(t)$ is defined pointwise with full probability but $E(|x(t)|)$ is infinite hence $E(x(t))$ is undefined. – Did Jun 18 '12 at 9:24
Actually for $\psi = \omega$ the solution is $$x(t) = C_1 \sin \omega t + C_2 \omega t - \frac{t \cos \omega t}{2 \omega}$$ so it's well defined. – qoqosz Jun 18 '12 at 11:02
@qoqosz: Typo---you omitted "cos" in the second term. – John Bentin Oct 5 '13 at 22:13
Suppose that $E$ is any equation, depending on a parameter $\psi\geqslant 0$, with a unique solution set $S_\psi$ for each value of psi. ($E$ does not need to be a differential equation.) What happens when a probability distribution is assigned to the possible values of $\psi$? Well, all you can say is that $S_\psi$ has the same distribution as $\psi$. (Continued...) – John Bentin Oct 6 '13 at 19:07
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. – John Bentin Oct 6 '13 at 19:18