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Given the Exact Differential $$M(x,y) \, dx + N(x,y) \, dy = 0\tag 1$$

I noticed that at least in Mathematica, the solution $\phi(x,y)$ can be found via solving the following differential equation from manipulation of $(1)$.

$$ \frac{dy}{dx} \equiv \frac{-M(x,y)}{N(x,y)} $$ I'm Standard Form, dropping $(x,y)$, $$ \frac{dy}{dx} + \frac{M}{N} \equiv 0$$ In Mathematica:

DSolve[y'[x]== -m[x,y[x]]/n[x,y[x]],y[x],x]

Is this commonly known, used, and if so, what is it called? Thank you all very much. I am starting to feel as though this should be in the Mathematica SE. If others believe so, please vote to move.

Michael Hardy
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Shinaolord
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  • What you say is of course true, however the right hand side is not necessarily easy to integrate. Try a simple example such as $N=xy$ and $M=x+y^2$. – Paul Sundheim Oct 03 '14 at 14:00
  • Yes I understand that, but, for example, after checking a hand solution via standard methods typing this into Mathematica will give you an easy way to check the solution to the differential, as Mathematica can't solve a difference of equation in the form stated in [1] above – Shinaolord Oct 03 '14 at 14:01
  • Is this true? My comment above didn't have a question my apologies – Shinaolord Oct 03 '14 at 14:02
  • I could see that Mathematica users (and perhaps other CAS users) might find it useful, but there is no name for it even though it is well known. – Paul Sundheim Oct 03 '14 at 14:12
  • Okay thank you @PaulSundheim, want to turn your comment into an answer? It hits the nail on the head perfectly. – Shinaolord Oct 03 '14 at 14:14
  • Ok. Done. See below. Thanks – Paul Sundheim Oct 03 '14 at 14:15

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I could see that Mathematica users (and perhaps other CAS users) might find it useful, but there is no name for it even though it is well known.

Paul Sundheim
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