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

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  • $\begingroup$ 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 ? $\endgroup$ – AgentS Jun 23 '18 at 17:44
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    $\begingroup$ Think about $x(y)$ instead of $y(x)$ $\endgroup$ – Claude Leibovici Jun 23 '18 at 17:48
  • $\begingroup$ @rsadhvika I've got this equation from my teacher he said that it converges to Riccati's equation $\endgroup$ – Павел Якимов Jun 23 '18 at 17:53
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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: $$ z' = z^2 - x^2. $$

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