Method of undetermined coefficients

Choose a finite trial family of polynomial-exponential or trigonometric functions for a particular solution, substitute it into a constant-coefficient linear equation, and solve for its coefficients. If the forcing exponent is a characteristic root of multiplicity $m$, multiply the ordinary trial by $t^m$ to leave the homogeneous kernel. The method is efficient for forcing families preserved by differentiation.

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 $x$.

For a constant-coefficient linear differential equation $P(D)y=e^{rx}$, where $D$ denotes differentiation, a simple root $P(r)=0$ forces a resonant trial. Factor $P(z)=(z-r)Q(z)$; since $(D-r)(x{e}^{rx})=e^{rx}$, commuting the differential operators gives $P(D)(x{e}^{rx})=Q(r)e^{rx}=P^{\prime }(r)e^{rx}$. Thus the displayed expression is a particular solution. More generally, if the root has multiplicity $m$, the same factorization gives $P(D)(x^m{e}^{rx})=P^{(m)}(r)e^{rx}$, so $y_p=x^m{e}^{rx}/P^{(m)}(r)$. This is the extra polynomial factor required by resonance in a differential equation.