# How to solve differential equation?

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

• First, why do you think it can be solved by hand ? Only a few types of differential equations can be solved manually. Your differential equation seems extremely difficult to solve w/o a computer. So may I ask where you got it from ? – AgentS Jun 23 '18 at 17:44
• Think about $x(y)$ instead of $y(x)$ – Claude Leibovici Jun 23 '18 at 17:48
• @rsadhvika I've got this equation from my teacher he said that it converges to Riccati's equation – Павел Якимов Jun 23 '18 at 17:53

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: $$z' = z^2 - x^2.$$