Difference between revisions of "Exact differential equation"
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To find the solution of this equation, we assume that the solution is φ = 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 φ = 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.) | ||
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| − | To solve the expression for <math>y | + | 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 φ 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 solve the expression for <math>y</math>, use the quadratic formula. | ||
[[Category:Calculus]] | [[Category:Calculus]] | ||
[[Category:Differential Equations]] | [[Category:Differential Equations]] | ||
Revision as of 18:55, 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:
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.
