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.

E 4.4 Exercises