Exact differential equation

Definition

Given a simply connected and open subset D of R² and two functions I and J which are continuous on D then an implicit first-order ordinary differential equation of the form

is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that

and

The nomenclature of "exact differential equation" refers to the exact derivative of a function. For a function F ( x 0 , x 1 , . . . , x n − 1 , x n ) , the exact or total derivative with respect to x 0 is given by

Example

The function F : R 2 → R given by

is a potential function for the differential equation

Existence of potential functions

In physical applications the functions I and J are usually not only continuous but even continuously differentiable. Schwarz's Theorem then provides us with a necessary criterion for the existence of a potential function. For differential equations defined on simply connected sets the criterion is even sufficient and we get the following theorem:

Given a differential equation of the form (for example, when F has zero slope in the x and y direction at F(x,y) ):

with I and J continuously differentiable on a simply connected and open subset D of R² then a potential function F exists if and only if

Solutions to exact differential equations

Given an Exact differential equation defined on some simply connected and open subset D of R² with potential function F then a differentiable function f with (x, f(x)) in D is a solution if and only if there exists real number c so that

For an initial value problem

we can locally find a potential function by

Solving

for y, where c is a real number, we can then construct all solutions.