Why is it that in order to solve an exact differential equation, the total differential must be zero? In other words, the following function must be equal to a constant

$ \phi (x,y) = c $

Then $ d \phi = \frac { \partial \phi}{ \partial x} dx + \frac { \partial \phi}{ \partial y} dy = 0$

Then a differential equation of the form $ M dx + N dy = 0 $ can be solved if $ \frac { \partial M}{ \partial y} = \frac { \partial N}{ \partial x}$

Then its called an 'exact differential equation'

But why in order to do this, there must be a condition where $ \phi (x,y) = c $ ?

Why can't it be something else? Thanks


The idea is to represent your solutions as level curves of a certain function $\Phi$. If that is the case, you should have $d\Phi=0$ along integral curves and the curves themselves will have the form of level curves $\Phi(x,y)=const.$


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