Method of Undetermined Coefficients

Definition

To solve $y^{\prime \prime }+a{y}^{\prime }+by=r(x)$, we have to solve corresponding homogeneous ODE $y^{\prime \prime }+a{y}^{\prime }+by=0$ and find a particular solution $y_p$

Table

Terms in $r(x)$	Choice for $y_p(x)$
$k{e}^{\gamma x}$	$C{e}^{\gamma x}$
k x^n,\quad _{n=0, 1, 2}	$K_n{x}^n+K_{n-1}{x}^{n-1}+\cdots +K_{1}x+K_0$
$k\cos \omega x$ $k\sin \omega x$	$K\cos \omega x+M\sin \omega x$
$k{e}^{\alpha x}\cos \omega x$ $k{e}^{\alpha x}\sin \omega x$	$e^{\alpha x}(K\cos \omega x+M\sin \omega x)$

Choice Rules

1. Basic Rule: If $r(x)$ belongs a column in the left, choose corresponding right $y_p$
2. Modification Rule: If $y_p$ is a solution of a corresponding homogeneous ODE, multiply the selected $y_p$ by $x$ or $x^2$
3. Sum Rule: If $r(x)$ is a sum of several functions, $y_p$ is selected as the sum of functions corresponding to each.