How to solve differential equation?

Question

How do we solve the differential equation ? $$\frac{dy}{dx} + 2x\sin(y) = 2.$$ I have no ideas for a solution.

Answer 1

Substitution: $$ t(x) = tg(\frac{y}{2}), $$ $$ y = 2 \ arctg(t), $$ $$ y' = \frac{2 t'}{1 + t^2}, $$ $$ sin(y) = \frac{2 t}{1 + t^2}. $$ We get: $$ \frac{2 t'}{1 + t^2} + 2x \frac{2 t}{1 + t^2} = 2$$ $$ t' + 2xt = 1 + t^2 $$ Substitution: $$ z = t - x $$ Riccati equation: $$ z' = z^2 - x^2. $$