Exact Equation

Given an equation, if there exists functions such that

Then the equation is an exact equation

The total derivative of a function can be given

Consider the equation

Assume can be given as

By taking the Partial Derivatives, we can see that is equal to the original equation

By equating the partial to we can see that is constant

Rather than finding and to determine exactness, the following process can be used

Let

Then take the complementary derivative of both equation

Assume is continuous and differentiable
If this condition is satisfied then the equation is exact

To determine exactness