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 can re-write a different form of this equation by substituting <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. This yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.
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To find the solution of this equation, we assume that the solution is &phi; = constant. We can re-write a different form of this equation by substituting <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. This yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>.
  
to find &phi;, we integrate M with respect to t and N with respect to y. This will give us two different equations. To find &phi; , we  
+
To find &phi;, we integrate M with respect to t and N with respect to y. This will give us two different equations. To find &phi; , we  
  
 
Go through the example to find &phi; by integrating, then check that
 
Go through the example to find &phi; by integrating, then check that

Revision as of 13:47, 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

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 can re-write a different form of this equation by substituting and . This yields .

To find φ, we integrate M with respect to t and N with respect to y. This will give us two different equations. To find φ , we

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

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.