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Differential Equations

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Moderate

IVP with a Given Derivative

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An unknown function i$y=y(x)$ satisfies:

$$(y\cos x+2xe^y+x)dx+(\sin x+x^2e^y-e^y)dy=0$$

If $y'(0)=e^{-1}$, which of the following is the solution curve?

A

$y\sin x+(x^2-1)e^y+\frac{x^2}{2}+e=0$

B

$2y\sin x+(2x^2-1)e^y+\frac{x^2}{2}+e=0$

C

$y\sin x+(x^2-1)e^y+\frac{x^2}{2}=-e^{e^{-1}}$

D

$2y\sin x+(2x^2-1)e^y+\frac{x^2}{2}=-e^{e^{-1}}$