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 differential 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

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