Exact differential equation

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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:

(1): 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 assume that 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. To get the main function φ 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.