Method of Undetermined Coefficients

Proof Technique
The method of undetermined coefficients is a technique for solving a nonhomogeneous linear second order ODE with constant coefficients:
 * $(1): \quad y'' + p y' + q y = R \left({x}\right)$

where $R \left({x}\right)$ is one of the following types of expression:
 * an exponential
 * a sine or a cosine
 * a polynomial

or a combination of such real functions.

Method and Proof
Let $y_g \left({x}\right)$ be the general solution to:
 * $y'' + p y' + q y = 0$

From Solution of Constant Coefficient Homogeneous LSOODE, $y_g \left({x}\right)$ can be found systematically.

Let $y_p \left({x}\right)$ be a particular solution to $(1)$.

Then from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
 * $y_g \left({x}\right) + y_p \left({x}\right)$

is the general solution to $(1)$.

It remains to find $y_p \left({x}\right)$.