Linear Second Order ODE/y'' + y = cosecant x

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

has the general solution:
 * $y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x$

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

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:
 * $y_g = C_1 \sin x + C_2 \cos x$

It remains to find a particular solution $y_p$ to $(1)$.

Expressing $y_g$ in the form:
 * $y_g = C_1 y_1 \left({x}\right) + C_2 y_2 \left({x}\right)$

we have:

By the Method of Variation of Parameters, we have that:


 * $y_p = v_1 y_1 + v_2 y_2$

where:

where $W \left({y_1, y_2}\right)$ is the Wronskian of $y_1$ and $y_2$.

We have that:

Hence:

It follows that:
 * $y_p = \sin x \ln \left({\sin x}\right) - x \cos x$

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


 * $y = y_g + y_p = C_1 \sin x + C_2 \cos x + \sin x \ln \left({\sin x}\right) - x \cos x$

is the general solution to $(1)$.