# Help in solving an inexact differential equation

#### vela

Staff Emeritus
Homework Helper
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

Homework Helper
Gold Member
2018 Award
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.

"Help in solving an inexact differential equation"

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