ODE/(D^4 - 1) y = sin x

Theorem
The second order ODE:
 * $(1): \quad \paren {D^4 - 1} y' = \sin x$

has a general solution:
 * $y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$

Proof
First we solve the reduced equation of $(1)$:
 * $(2): \quad \paren {D^4 - 1} y' = 0$

The auxiliary equation of $(1)$ is:
 * $(3): \quad: m^4 - 1 = 0$

From Complex $4$th Roots of Unity, the roots of $(2)$ are:
 * $m_1 = 1$
 * $m_2 = i$
 * $m_3 = -1$
 * $m_4 = -i$

So from Solution of Constant Coefficient Linear nth Order ODE, the general solution of $(2)$ is:
 * $y_g = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x$

Because $\sin x$ is already a solution of $(2)$, we try:
 * $y = A x e^{i x}$

with a view to taking the real part in due course.

It follows that
 * $-4 i A - 1$

and so:
 * $A = -\dfrac 1 4$

Thus a particular solution to $(1)$ can be given as:

and the general solution is:
 * $y = C_1 e^x + C_2 e^{-x} + C_3 \sin x + C_4 \cos x + \dfrac {x \cos x} 4$