Difference between revisions of "Exact differential equation"
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Revision as of 13:04, 2 August 2010
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:
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or Failed to parse (lexing error):
The solution is φ = constant, where φ is determined 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.)
or
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.