# 4.4 | Nonhomogeneous Equations: The Method of Undetermined Coefficients

Method of Undetermined Coefficients To find a particular solution to the differential equation
$$ay^{\prime \prime} +by’ + cy = Ct^m e^{rt},$$
where $$m$$ is a nonnegative integer, use the form
$$y_p(t) = t^s (A_m t^m + \cdots + A_1 t + A_0) e^{rt},$$
with

1. $$s = 0$$ if $$r$$ is a not a root of the associated auxiliary equation;
2. $$s = 1$$ if $$r$$ is a simple root of the associated auxiliary equation; and
3. $$s = 2$$ if $$r$$ is a double root of the associated auxiliary equation.

To find a particular solution to the differential equation
$$ay^{\prime \prime} + by’ + cy = \begin{cases} Ct^me^{\alpha t} \cos \beta t \\ \text{or} \\ Ct^me^{\alpha t} \sin \beta t \end{cases}$$
for $$\beta \not = 0,$$ use the form
\begin{aligned} y_p(t) & = t^s(A_m t^m + \cdots + A_1 t + A_0) e^{\alpha t} \cos \beta t \\ & \quad + t^s(B_m t^m + \cdots + B_1 t + B_0) e^{\alpha t} \sin \beta t \end{aligned}
with

1. $$s = 0$$ if $$\alpha + i \beta$$ is a not a root of the associated auxiliary equation;
2. $$s = 1$$ if $$\alpha + i \beta$$ is a root of the associated auxiliary equation.