Last modified on 2 August 2010, at 13:40

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:


To find the solution of this equation, we assume that the solution is φ = constant. This means that: and

φ is found by integrating M and N:

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


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

(That's the condition for "exactness" of the differential form M dt + N dy.)

where .

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.