# Why do exact differential equations have to equal zero?

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

## 1 Answer

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 $$\Phi(x,y)=const.$$