# Exact differential equation

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In mathematics, an **exact differential equation** or **total differential equation** is a certain kind of ordinary differential equation which is widely used in physics and engineering.

## Definition

Given a simply connected and open subset *D* of **R**^{2} 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 , the exact or total derivative with respect to is given by

### Example

The function

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 **R**^{2} 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 **R**^{2} 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.

## See also

## References

- Boyce, William E.; DiPrima, Richard C. (1986).
*Elementary Differential Equations*(4th ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-07894-8