 Complex forcing method for a real differential equation

For a real linear differential operator $L$, one has $L(\mathrm{Im}u)=\mathrm{Im}(Lu)$. A forcing term written as the imaginary part of a polynomial times $e^{i\omega x}$ can therefore be handled by solving one complex equation, then taking the imaginary part. For $L=d^2/d{x}^2+a$, substituting $u=p(x)e^{i\omega x}$ gives $p^{\prime \prime }+2i\omega p^{\prime }+(a-{\omega }^2)p$. If the forcing frequency is resonant, the polynomial ansatz must include the appropriate additional power of $x$.