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Help in solving an inexact differential equation

vela

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I guess the two forms must be equivalent, but that is far from obvious.
It's not too bad, actually. In terms of your variables, LCKurtz's ##u## and ##v## are
\begin{align*}
u &= X + Y \\
v &= -X + Y
\end{align*} which we can interpret as a rotation by ##45^\circ## and a rescaling by ##\sqrt 2##. Interpreting the terms in the two solutions geometrically, it makes sense that the two are equivalent.
 

haruspex

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It's not too bad, actually. In terms of your variables, LCKurtz's ##u## and ##v## are
\begin{align*}
u &= X + Y \\
v &= -X + Y
\end{align*} which we can interpret as a rotation by ##45^\circ## and a rescaling by ##\sqrt 2##. Interpreting the terms in the two solutions geometrically, it makes sense that the two are equivalent.
No, I meant after substituting back the original x and y. I agree the two arctan forms:
##\arctan(\frac{-x+y+6}{x+y+4})## and ##\arctan(\frac{x-1}{y+5})##
do turn out to differ by a constant, but it is surprising.
 

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