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Exact Differential Equation

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A differential equation of the form

$\displaystyle q(x,y) y^\prime + p(x,y) = 0 , \ q(x,y) \neq 0 $

for $ y = y(x)$ is called exact if exists a function $ F(x,y)$ with

$\displaystyle p=F_x,\ q = F_y \Leftrightarrow (p,q)^{\operatorname t}= \operatorname{grad} F \,, $

If $ p$ and $ q$ are continuous differentiable functions and are defined in a simply connected region, e.g. in an open rectangle or an open circle or in the entire plane, then the existence of $ F$ is equivalent to

$\displaystyle q_x = p_y .$

If the regio is not simply connected then the condition $ q_x = p_y$ is necessary for the existence of $ F .$

The solutions of an exact differential equation are implicitely given by

$\displaystyle F(x,y) = c \,, $

where $ c$ is a constant which may fixed by certain initial values.

Quite often the differential equation is also written in the form

$\displaystyle p(x,y) dx + q(x,y) dy = 0 \,. $

()

Example:

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  automatically generated 9/13/2005