Method of Undetermined Coefficients/Sine and Cosine/Particular Solution/i b is Root of Auxiliary Equation/Trigonometric Form

Proof Technique
Consider the nonhomogeneous linear second order ODE with constant coefficients:
 * $(1): \quad y'' + b^2 y = \alpha \sin b x + \beta \cos b x$

The Method of Undetermined Coefficients can be used to find a particular solution to $(1)$ in the following manner.

Method and Proof
Let $\map {y_g} x$ be the general solution to:
 * $y'' + b^2 y = 0$

From General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:
 * $\map {y_g} x + \map {y_p} x$

is the general solution to $(1)$.

It remains to find $\map {y_p} x$.

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

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

We have:

Inserting into $(1)$:

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

Hence:
 * $y_p = \dfrac {\beta x \sin b x} {2 b} - \dfrac {\alpha x \cos b x} {2 b}$