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:
(we will call this equation 1)
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 assume the substitution and . (If we substitute M and N back into (1), it yields , which makes sense.)
To find , manipulate the substitutions of M and N to get and . Integrate both sides. This will give us and . To get , write the sum of each term found in each equation. For terms that appear in both equations, only write them once.
To solve the expression for , use the quadratic formula.