Method of undermined coefficients.

These notes are set so that you get to prove the main results by solving smaller problems that when put together give the big result. The answers to the problems are in the videos. You will get the most out of these notes if you do (or try) the problems before looking at the videos.

Method of undetermined coefficients for constant coefficient equations.

We have seen that if $y_1$ and $y_2$ are linearly independent solutions to the linear homogeneous differential equation $$ L(y) := ay''+by'+cy=0 $$ where $a$, $b$, and $c$ are constants then the general solution to this equation is $$ y_c = c_1y_1 + c_2 . $$ Then if we have a particular solution $y$ to the inhomogeneous equation $$ L(y_p) = ay_p'' + by_p' + cy_p =f(x) $$ then the general solution to the inhomogeneous is $$ y = y_p + y_c. $$

For some choices of $f(x)$ we can make a educated guess for $y_p$ and reduce the problem to solve for some constants. Here is a list of some $f(x)$ and the corresponding choice of $y_p$.

$f(x)$	$y_p$
$C$	$A$
$C_1x+C_0$	$AX+B$
$C_2x^2+C_1x+C_0$	$AX^2+Bx+C$
$C_nx^n + C_{n-1}x^{n-1} + \cdots + C_1x+C_0$	$A_nx^n + A_{n-1}x^{n-1} + \cdots + A_1x+A_0$
$Ce^{ax}$	$A e^{ax}$
$C_0 \cos(bx) + C_1 \sin(bx)$	$A \cos(bx) + B \sin(bx)$
$e^{ax}\big(C_0 \cos(bx) + C_1 \sin(bx)\big)$	$e^{ax}\big(A \cos(bx) + B \sin(bx)\big)$

Problem: Find the general solution to $$ y'' - 9 y = 24 \cos(3x). $$

Solution.

One thing that can go wrong with this method is that if $f(x)$ is a solution to the equation. There is a modification of the choice of $y_p$ that often works, replace the initial choice of your guess $y_p$ with $ x y_p$. That is take your original guess and multiply it by $x$. Here is an example.

Problem: Find the general solution to $$ y ''- y' - 6y = 12 + 18 e^{-2x}. $$

Solution.