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 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:$frac\{mathrm\{d\}F\}\{mathrm\{d\}x\_0\}=frac\{partial\; F\}\{partial\; x\_0\}+sum\_\{i=1\}^\{n\}frac\{partial\; F\}\{partial\; x\_i\}frac\{mathrm\{d\}x\_i\}\{mathrm\{d\}x\_0\}.$

Examples

The function :$F(x,y)\; :=\; frac\{1\}\{2\}(x^2\; +\; y^2)$is a potential function for the differential equation:$xx\text{'}\; +\; yy\text{'}\; =\; 0.,$

Existence of potential functions

In physical applications the functions "I" and "J" are usually not only continuous but even continuously differentiable. Schwarz's Theorem (also known as Clairaut'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: $I(x,\; y),\; dx\; +\; J(x,\; y),\; dy\; =\; 0,\; ,!$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 :$frac\{partial\; I\}\{partial\; y\}(x,\; y)\; =\; frac\{partial\; J\}\{partial\; x\}(x,\; y).$

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:$F(x,y)\; =\; int\_\{x\_0\}^x\; I(t,y\_0)\; dt\; +\; int\_\{y\_0\}^y\; J(x,t)\; dt.$

Solving :$F(x,y)\; =\; c,$for "y", where "c" is a real number, we can then construct all solutions.

See also

* exact differential
* total derivative