Difference between revisions of "Exact differential equation"
| Line 4: | Line 4: | ||
<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 φ = constant. | + | To find the solution of this equation, we assume that the solution is φ = 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> |
| + | |||
φ is found by integrating M and N: | φ 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 18:39, 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
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.
