ODE4.5

Find the general solution of the equation $u''+\omega^2 u=f(t)\,$ where $\omega\isin\mathbb{R}\,$ .

Two linearly independent solutions to the homogeneous equation are $u^1=cos \omega t, u^2=\sin \omega t\,$ .

$X=\begin{bmatrix} cos\omega t & \sin \omega t \\ -\omega \sin \omega t & \omega \cos \omega t \end{bmatrix}\,$

$W = \omega \cos^2\omega t + \omega \sin^2 \omega t = \omega\,$

From ODE theory, the particular solution is

$u_p(t)=\int_a^t \frac{f(s)}{W(s)}\left[u^2(t)u^1(s)-u^1(t)u^2(s)\right] ds\,$

$= \frac{1}{\omega} \int_a^t f(s)\left[ \sin \omega t \cos \omega s - \cos\omega t \sin\omega s\right] ds\,$

$= \frac{1}{\omega}\int_a^t f(s)\sin \omega(t-s)ds\,$

So the general solution is

$u(t) = \cos\omega t + \sin\omega t + \frac{1}{\omega} \int_a^t f(s) \sin \omega (t-s) ds\,$