Solving the partial differential equation $f_{xy}=\frac{-e^x\sin{y}+e^y\sin{x}}{\phantom{-}e^x\cos{y}+e^y\cos{x}}$

Asked Dec 22 '19 at 14:20

Active Dec 22 '19 at 16:53

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How could we evaluate this indefinite double integral in the variables $x$ and $y$? $$\iint \frac{-e^x\sin{y}+e^y\sin{x}}{\phantom{-}e^x\cos{y}+e^y\cos{x}}dxdy$$

The argument of the double integral must be the derivative of a $y=f(x)$ function; being this quite strange, I think that in the double integration the $y$ could simplify.

For what a double integral is, see here.

In other words, what are the solutions $f$ to the following PDE?

$$f_{xy}=\frac{-e^x\sin{y}+e^y\sin{x}}{\phantom{-}e^x\cos{y}+e^y\cos{x}}$$

edited Jun 12 '20 at 10:38

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asked Dec 22 '19 at 14:20

Shootforthemoon

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What does a double indefinite integral mean? – GEdgar Dec 22 '19 at 14:33
it is the antiderivative of a two variable function – Shootforthemoon Dec 22 '19 at 14:34
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What does "antiderivative of a two variable function" mean? – GEdgar Dec 22 '19 at 14:34
The argument of the double integral must be the derivative of a y=f(x) function – Shootforthemoon Dec 22 '19 at 14:38
Since it is quite strange, I think that in the double integration the y could simplify – Shootforthemoon Dec 22 '19 at 14:39
https://math.stackexchange.com/questions/704186/indefinite-double-integral?rq=1 – Shootforthemoon Dec 22 '19 at 14:42
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Since this is an unknown concept, it would be best if that link were included in the question! – GEdgar Dec 22 '19 at 14:47
alright, thanks for your suggestion – Shootforthemoon Dec 22 '19 at 14:53
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Instead of phrasing the question in terms of rather nonstandard concept, why not simply ask for the solutions $f$ of the PDE $f_{xy}=\dots$? – Hans Lundmark Dec 22 '19 at 16:26
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I would try switching the coordinates to $u=x+iy, v=x-iy$. – Lee Dec 23 '19 at 00:40

Solving the partial differential equation $f_{xy}=\frac{-e^x\sin{y}+e^y\sin{x}}{\phantom{-}e^x\cos{y}+e^y\cos{x}}$

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