Difference between revisions of "Exact differential equation"

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<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
 
<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
  
To find the solution of this equation, we assume that the solution is &phi; = constant. We assume that <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. (If we substitute M and N back into (1), it yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.)
+
To find the solution of this equation, we assume that the solution is &phi; = constant. We assume the substitution <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. (If we substitute M and N back into (1), it yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.)
  
  

Revision as of 13:56, 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. 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.