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.

1 Definition
- 1.1 Examples
2 Existence of potential functions
3 Solutions to exact differential equations
4 See also

[edit] 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

$I(x, y)\, \mathrm{d}x + J(x, y)\, \mathrm{d}y = 0, \,\!$

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

$\frac{\partial F}{\partial x}(x, y) = I$

and

$\frac{\partial F}{\partial y}(x, y) = J.$

The nomenclature of "exact differential equation" refers to the exact derivative (or total 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

$\frac{\mathrm{d}F}{\mathrm{d}x_0}=\frac{\partial F}{\partial x_0}+\sum_{i=1}^{n}\frac{\partial F}{\partial x_i}\frac{\mathrm{d}x_i}{\mathrm{d}x_0}.$

[edit] Examples

The function

$F(x,y) := \frac{1}{2}(x^2 + y^2)$

is a potential function for the differential equation

$xx' + yy' = 0.\,$

[edit] Existence of potential functions

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

Given a differential equation of the form

$I(x, y)\, dx + J(x, y)\, dy = 0, \,\!$

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

$\frac{\partial I}{\partial y}(x, y) = \frac{\partial J}{\partial x}(x, y).$

[edit] 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