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.
 
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:
 
Suppose you are given an equation of the form:
  
:(1)<math>M(t,y) + N(t,y)y' = 0\,</math>  or  <math>M(t,y) dt + N(t,y) dy = 0\,</math>
+
:<math>M(t,y) + N(t,y)y' = 0\,</math>  or  <math>M(t,y) dt + N(t,y) dy = 0\,</math> (equation 1)
  
  

Revision as of 13:55, 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:

or (equation 1)


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.