Difference between revisions of "Exact differential equation"
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:(1)<math>M(t,y) + N(t,y)y' = 0\,</math> or <math>M(t,y) dt + N(t,y) dy = 0\,</math> | :(1)<math>M(t,y) + N(t,y)y' = 0\,</math> or <math>M(t,y) dt + N(t,y) dy = 0\,</math> | ||
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Before we begin solving it, we must first check that the equation is exact. This means that: | Before we begin solving it, we must first check that the equation is exact. This means that: | ||
Revision as of 18:54, August 2, 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:
- (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.
