Difference between revisions of "Exact differential equation"

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<math>M(t,y) + N(t,y)y' = 0\,</math>
 
<math>M(t,y) + N(t,y)y' = 0\,</math>
 +
 
or
 
or
 +
 
<math>M(t,y) dt + N(t,y) dy = 0\,</math>
 
<math>M(t,y) dt + N(t,y) dy = 0\,</math>
  
To find the solution of this equation, we assume that the solution is &phi; = constant.
+
To find the solution of this equation, we assume that the solution is &phi; = constant. This means that:
This means that <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>
+
<math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>
 +
 
 
&phi; is found by integrating M and N:
 
&phi; is found by integrating M and N:
 
:<math>\phi(t, y) = \int_0^t M(s, 0) ds + \int_0^y N(t, s) ds</math>
 
:<math>\phi(t, y) = \int_0^t M(s, 0) ds + \int_0^y N(t, s) ds</math>

Revision as of 13:39, 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

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.