# Difference between revisions of "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:



or



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



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.