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Find all solutions of the differential equation:

$$ (dy/dx)^2 + e^y +cos(x) + 3 = 0 $$

I'm having trouble figuring out where to start. Examples I've seen have polynomial terms instead of exponential and trig terms.

Thank you.

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  • $\begingroup$ Is $y$ a real function? At least one of the first 3 terms has to be smaller than or equal $-1$. What does that say about the domain of the equation? $\endgroup$ – Lutz Lehmann Feb 3 '19 at 0:32
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Since $e^y \ge 0$ and $\cos x \ge -1$ we have

$$ (y')^2 + e^y + \cos x + 3 \ge 2 $$

So there is no solution.

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