Difference between revisions of "Exact differential equation"

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To find <math>y</math>, manipulate the substitutions of M and N to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides. To get the main function &phi; write the sum of each term found in each equation. For terms that appear in both equations, only write them once.
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To find <math>y</math>, manipulate the substitutions of M and N to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides. This will give us <math>\phi(t)\,</math> and <math>\phi(y)\,</math>. To get <math>\phi(t, y)\,</math>, write the sum of each term found in each equation. For terms that appear in both equations, only write them once.
  
  

Revision as of 13:58, 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

(we will call this 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 the substitution 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. This will give us and . To get , 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.