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.

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

$F(x,y)=\int _{{x_{0}}}^{x}I(t,y_{0}){\mathrm {d}}t+\int _{{y_{0}}}^{y}\left[J(x,t)-\int _{{x_{0}}}^{{x}}{\frac {\partial I}{\partial t}}(u,t)\,{\mathrm {d}}u\,\right]{\mathrm {d}}t.$

Solving

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

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

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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