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} = I

and

\frac{\partial F}{\partial 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 $F:\mathbb{R}^{2}\to\mathbb{R}$ given by

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

is a potential function for the differential equation

x\,\mathrm{d}x + y\,\mathrm{d}y = 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}J(x,t)\mathrm {d} t=\int _{x_{0}}^{x}I(t,y_{0})\mathrm {d} t+\int _{y_{0}}^{y}\left[J(x_{0},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.

References

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

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