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:
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.)
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.