exact differential equation

Let $R$ be a region in ${ℝ}^2$ and let the functions  $X:R\to ℝ$,  $Y:R\to ℝ$ have continuous partial derivatives in $R$.  The first order differential equation

$$X(x,y)+Y(x,y)\frac{dy}{dx}=\mathrm{\hspace{0.33em}0}$$

or

$X(x,y)dx+Y(x,y)dy=\mathrm{\hspace{0.33em}0}$ (1)

is called an exact differential equation, if the condition

${\displaystyle \frac{\partial X}{\partial y}}={\displaystyle \frac{\partial Y}{\partial x}}$ (2)

is true in $R$.

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

$$df(x,y)=\mathrm{\hspace{0.33em}0},$$

whence its general integral is

$$f(x,y)=C.$$

The solution function $f$ can be calculated as the line integral

$f(x,y):={\displaystyle {\int }_{P_0}^P}[X(x,y)dx+Y(x,y)dy]$ (3)

along any curve $\gamma$ connecting an arbitrarily chosen point  $P_0=(x_0,y_0)$  and the point  $P=(x,y)$  in the region $R$ (the integrating factor is now $\equiv 1$).

Example.  Solve the differential equation

$$\frac{2x}{y^3}dx+\frac{y^2-3x^2}{y^4}dy=\mathrm{\hspace{0.33em}0}.$$

This equation is exact, since

$$\frac{\partial }{\partial y}\frac{2x}{y^3}=-\frac{6x}{y^4}=\frac{\partial }{\partial x}\frac{y^2-3x^2}{y^4}.$$

If we use as the integrating way the broken line from  $(0,\mathrm{\hspace{0.17em}1})$  to  $(x,\mathrm{\hspace{0.17em}1})$  and from this to  $(x,y)$,  the integral (3) is simply

$${\int }_0^x\frac{2x}{1^3}𝑑x+{\int }_1^y\frac{y^2-3x^2}{y^4}𝑑y=\frac{x^2}{y^3}-\frac{1}{y}+1=x^2-\frac{1}{y}+\frac{x^2}{y^3}+1-x^2=\frac{x^2}{y^3}-\frac{1}{y}+1.$$

Thus we have the general integral

$$\frac{x^2}{y^3}-\frac{1}{y}=C$$

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