Exact differential equation

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 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 an 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 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

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

Example

The function

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

is a potential function for the differential equation

$xdx + ydy = 0.\,$

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) ):

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

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

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

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

$F(x, f(x)) = c.\,$

For an initial value problem

$y(x_0) = y_0\,$

we can locally find a potential function by

$F(x,y) = \int_{x_0}^x I(t,y_0) \mathrm{d}t + \int_{y_0}^y \left[ J(x,t) - \int_{x_0}^{x} \frac{\partial I}{\partial t}(u, t)\, \mathrm{d}u\,\right]\mathrm{d}t.$

Solving

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

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