Exact differential equation

An exact differential equation is a differential equation that can be solved in the following manner.

Suppose you are given an equation of the form:

or

Before we begin solving it, we must first check that the equation is exact. This means that:

To find the solution of this equation, we assume that the solution is φ = constant. We can re-write a different form of this equation by substituting and . This yields .

To find φ, we integrate M with respect to t and N with respect to y. This will give us two different equations. To find φ , we

Go through the example to find φ by integrating, then check that

and

and that any function φ = some constant, when turned into the corresponding dy/dt, satisfies the original equation. Be sure to emphasize that one must check first that

To find , first set and . Then manipulate to get and . Integrate both sides, compare the results for , and combine the terms into one equation (for terms that show up in both expressions, only write once in the combined expression.) To solve the expression for , plug into the quadratic formula.