Schiffler point

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 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

$F(x,y)=\underset{x_0}{\overset{x}{\int }}I(t,y_0)\mathrm{d}t+\underset{y_0}{\overset{y}{\int }}[J(x,t)-\underset{x_0}{\overset{x}{\int }}\frac{\partial I}{\partial t}(u,t)\mathrm{d}u]\mathrm{d}t.$

Solving

$F(x,y)=c$

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

See also

• Exact differential

References

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