How do u solve y` - y cot x = 2x - x^2 cot x

Question

I tried solving this ODE but I couldn't proceed beyond y/ sin (x) = int(2x csc (x) - (x^2) cot (x) csc (x) dx) I'm not really sure that my assumption that this could be solved as a linear ODE is really valid. ... however if it is then I stopped because I can't tell how to integrate x csc x

Answer 1

Solve y` - y cot x = 2x - x^2 cot x

Rearrange the DE.

y' +(x^2 - y)cot(x) = 2x

(x^2 - y)cot(x) = 2x - y'

Let g(x) = x^2 - y, then g'(x) = 2x - y'

So,

g(x).cot(x) = 2x - y' = g'(x)

Or,

g' - g.cot(x) = 0

The solution for this is,

g(x) = A.sin(x)

Substituting back for g(x) = A.sin(x)

x^2 - y = A.sin(x)

y(x) = x^2 - A.sin(x)