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[parent] exact differential equation (Result)

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

$\displaystyle X(x,\,y)+Y(x,\,y)\frac{dy}{dx} = 0$
or
$\displaystyle X(x,\,y)dx+Y(x,\,y)dy = 0$ (1)

is called an exact differential equation, if the condition
$\displaystyle \frac{\partial X}{\partial y} = \frac{\partial Y}{\partial x}$
is true in $ R$.

Then there is a function $ f\!: R \to \mathbb{R}$ such that the equation (1) has the form

$\displaystyle d\,f(x,\,y) = 0,$
whence its general integral is
$\displaystyle f(x,\,y) = C.$

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

$\displaystyle f(x,\,y) := \int_{P_0}^P [X(x,\,y)\,dx+Y(x,\,y)\,dy]$ (2)

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

$\displaystyle \frac{2x}{y^3}\,dx+\frac{y^2-3x^2}{y^4}\,dy = 0.$
This equation is exact, since
$\displaystyle \frac{\partial}{\partial y}\frac{2x}{y^3} = -\frac{6x}{y^4} = \frac{\partial}{\partial x}\frac{y^2-3x^2}{y^4}.$
If we use as the integrating way the broken line from $ (0,\,1)$ to $ (x,\,1)$ and from this to $ (x,\,y)$, the integral (2) is simply
$\displaystyle \int_0^x\frac{2x}{1^3}\,dx+\!\int_1^y\frac{y^2-3x^2}{y^4}\,dy = \... ...}{y}+1 = x^2-\frac{1}{y}+\frac{x^2}{y^3}+1-x^2 = \frac{x^2}{y^3}-\frac{1}{y}+1.$
Thus we have the general integral
$\displaystyle \frac{x^2}{y^3}-\frac{1}{y} = C$
of the given differential equation.



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See Also: harmonic conjugate function

Also defines:  exact differential equation

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Cross-references: integral, broken line, integrating factor, point, curve, line integral, solution, general integral, equation, differential equation, first order, partial derivatives, continuous, functions, region
There is 1 reference to this entry.

This is version 6 of exact differential equation, born on 2008-06-02, modified 2008-10-02.
Object id is 10648, canonical name is ExactDifferentialEquation.
Accessed 635 times total.

Classification:
AMS MSC: 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
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