exact differential equation

Let R be a region in 2 and let the functions  X:R,  Y:R have continuousMathworldPlanetmath partial derivativesMathworldPlanetmath in R.  The first order differential equationMathworldPlanetmath

X(x,y)+Y(x,y)dydx= 0


X(x,y)dx+Y(x,y)dy= 0 (1)

is called an exact differential equation, if the condition

Xy=Yx (2)

is true in R.

By (2), the left hand side of (1) is the total differentialMathworldPlanetmath of a function, there is a function  f:R  such that the equation (1) reads

df(x,y)= 0,

whence its general integral is


The solution function f can be calculated as the line integral

f(x,y):=P0P[X(x,y)dx+Y(x,y)dy] (3)

along any curve γ connecting an arbitrarily chosen point  P0=(x0,y0)  and the point  P=(x,y)  in the region R (the integrating factorMathworldPlanetmath is now 1).

Example.  Solve the differential equation

2xy3dx+y2-3x2y4dy= 0.

This equation is exact, since


If we use as the integrating way the broken line from  (0, 1)  to  (x, 1)  and from this to  (x,y),  the integral (3) is simply


Thus we have the general integral


of the given differential equation.

Title exact differential equation
Canonical name ExactDifferentialEquation
Date of creation 2013-03-22 18:06:17
Last modified on 2013-03-22 18:06:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Result
Classification msc 34A05
Related topic HarmonicConjugateFunction
Related topic Differential
Related topic TotalDifferential
Defines exact differential equation