# Difference between revisions of "Exact differential equation"

(Created page with '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: <math>M y' + N = 0 \<…') |
|||

Line 3: | Line 3: | ||

Suppose you are given an equation of the form: | Suppose you are given an equation of the form: | ||

− | <math>M y | + | <math>M(t,y) + N(t,y)y' = 0\,</math> or <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, where φ is determined 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> | ||

Line 16: | Line 16: | ||

:<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math> | :<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math> | ||

(That's the condition for "exactness" of the differential form M dt + N dy.) | (That's the condition for "exactness" of the differential form M dt + N dy.) | ||

− | |||

− | |||

− | |||

## Revision as of 13:35, 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

To find the solution of this equation, we assume that the solution is φ = constant, where φ is determined 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.