Linear First Order ODE/y' + y cot x = 2 x cosec x

Theorem
The linear first order ODE:
 * $(1): \quad y' + y \cot x = 2 x \csc x$

has the general solution:
 * $y = x^2 \csc x + C \csc x$

Proof
$(1)$ is in the form:
 * $\dfrac {\d y} {\d x} + \map P x y = \map Q x$

where:
 * $\map P x = \cot x$

Thus:

Thus from Solution by Integrating Factor, $(1)$ can be rewritten as:
 * $\dfrac {\d} {\d x} y \sin x = 2 x$

and the general solution is:
 * $y \sin x = x^2 + C$

or:
 * $y = x^2 \csc x + C \csc x$