ODE4.5

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


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

So the fundamental matrix is

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

The Wronskian is

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\,


Main Page : Ordinary Differential Equations : Second order, linear, nonhomogeneous equations