Method of Undetermined Coefficients/Sine and Cosine

Proof Technique
Consider the nonhomogeneous linear second order ODE with constant coefficients:
 * $(1): \quad y'' + p y' + q y = R \left({x}\right)$

Let $R \left({x}\right)$ be a linear combination of sine and cosine:
 * $R \left({x}\right) = \alpha \sin b x + \beta \cos b x$

The Method of Undetermined Coefficients can be used to solve $(1)$ in the following manner.

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 $y_p \left({x}\right)$.

Let $R \left({x}\right) = \alpha \sin b x + \beta \cos b x$.

Consider the auxiliary equation to $(1)$:
 * $(2): \quad m^2 + p m + q = 0$

There are two cases to consider.


 * $b$ is not a root of $(2)$

Assume that there is a particular solution to $(1)$ of the form:
 * $y_p = A \sin b x + B \cos b x$

We have:

Inserting into $(1)$:

Hence $A$ and $B$ can be expressed in terms of $\alpha$ and $\beta$:

Hence:
 * $y_p = \dfrac{\alpha \left({q - b^2}\right) - \beta b p} {\left({q - b^2}\right)^2 + b^2 p^2} \sin b x + \dfrac{\beta \left({q - b^2}\right) - \alpha b p} {\left({q - b^2}\right)^2 + b^2 b^2} \cos b x$