Solve sin x dy/dx + y cos x = x sin x

sin x dy/dx + y cos x = x sin x

Recognise an exact derivative

The left side is d(y sin x)/dx. So

d(y sin x)/dx = x sin x

Integrate

y sin x = ∫ x sin x dx = −x cos x + ∫ cos x dx = −x cos x + sin x + c

y sin x = sin x − x cos x + c

or y = 1 − x cot x + c cosec x.

(Alternative check by the standard method: dividing by sin x gives y' + y cot x = x with IF = sin x, which leads to the same result.)