Used when the right hand side of a [[Ordinary Differential Equations (ODEs)#Non-homogeneous|non-homogeneous ODE]] is a sum or product of simple polynomial, exponential, and trigonometric functions. A guess is made about the form of the solution, which is then substituted in and coefficients are selected to satisfy the equation. | $f(t)$ | Prototype | | -------------------------- | -------------------------- | | $e^{at}$ | $Ae^{at}$ | | $\cos(at)$ | $A\cos(at)+B\sin(at)$ | | $\sin(at)$ | $A\cos(at)+B\sin(at)$ | | $at^n+bt^{n-1}+\dots+mt+n$ | $At^n+Bt^{n-1}+\dots+Mt+N$ | Products of functions can be represented by products of their prototypes, for example: | $f(t)$ | Prototype | | ---------------- | --------------------------------- | | $e^{at}\sin(bt)$ | $Ae^{at}\cos(bt)+Be^{at}\sin(bt)$ | | $t^2e^{at}$ | $At^2e^{at}+Bte^{at}+Ce^{at}$ |