Linear Second Order ODE/y'' + y = 2 cosine x

Theorem
The second order ODE:
 * $(1): \quad y'' + y = 2 \cos x$

has the general solution:
 * $y = C_1 \sin x + C_2 \cos x + x \cos x$

Proof
It can be seen that $(1)$ is a nonhomogeneous linear second order ODE with constant coefficients in the form:
 * $y'' + p y' + q y = R \left({x}\right)$

where:
 * $p = 0$
 * $q = 1$
 * $R \left({x}\right) = 2 \cos x$

First we establish the solution of the corresponding constant coefficient homogeneous linear second order ODE:
 * $(2): \quad y'' + y = 0$

From Second Order ODE: $y'' + y = 0$, this has the general solution:
 * $y_g = C_1 \sin x + C_2 \cos x$

We have that:
 * $R \left({x}\right) = 2 \cos x$

and it is noted that $2 \cos x$ is a particular solution of $(2)$.

So from the Method of Undetermined Coefficients for Polynomials:
 * $y_p = A x \sin x + B x \cos x$

where $A$ and $B$ are to be determined.

Hence:

Substituting into $(1)$:

So from General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution:


 * $y = y_g + y_p = C_1 \sin x + C_2 \cos x + x \sin x$