Definition:Exact Differential Equation

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Definition

Let a first order ordinary differential equation be expressible in this form:

$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

such that $M$ and $N$ are not homogeneous functions of the same degree.


However, suppose there happens to exist a function $\map f {x, y}$ such that:

$\dfrac {\partial f} {\partial x} = M, \dfrac {\partial f} {\partial y} = N$

such that the second partial derivatives of $f$ exist and are continuous.


Then the expression $M \rd x + N \rd y$ is called an exact differential, and the differential equation is called an exact differential equation.


Also presented as

An exact differential equation can also be presented as:

$\dfrac {\d y} {\d x} = -\dfrac {\map M {x, y} } {\map N {x, y} }$


or:

$\dfrac {\d y} {\d x} + \dfrac {\map M {x, y} } {\map N {x, y} } = 0$


or in differential form as:

$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$


or:

$\map M {x, y} \rd x = -\map N {x, y} \rd y$


all with the same conditions on $\map f {x, y}$, $\dfrac {\partial f} {\partial x}$ and $\dfrac {\partial f} {\partial y}$.


Also known as

An exact differential equation is usually referred to as just an exact equation.


Examples

$e^y \rd x + \paren {x e^y + 2 y} \rd y = 0$

is an exact differential equation with solution:

$x e^y + y^2 = C$


$\paren {x + \dfrac 2 y} \rd y + y \rd x = 0$

is an exact differential equation with solution:

$x y + 2 \ln y = C$


$\paren {y - x^3} \rd x + \paren {x + y^3} \rd y = 0$

is an exact differential equation with solution:

$4 x y - x^4 + y^4 = C$


$\paren {y + y \cos x y} \rd x + \paren {x + x \cos x y} \rd y = 0$

is an exact differential equation with solution:

$x y + \sin x y = C$


$\paren {\sin x \sin y - x e^y} \rd y = \paren {e^y + \cos x \cos y} \rd x$

is an exact differential equation with solution:

$\sin x \cos y + x e^y = C$


$-\dfrac 1 y \sin \dfrac x y \rd x + \dfrac x {y^2} \sin \dfrac x y \rd y = 0$

is an exact differential equation with solution:

$\dfrac x y = C$


$\paren {2 x y^3 + y \cos x} \rd x + \paren {3 x^2 y^2 + \sin x} \rd y = 0$

is an exact differential equation with solution:

$x^2 y^3 + y \sin x = C$


$\d x = \dfrac y {1 - x^2 y^2} \rd x + \dfrac x {1 - x^2 y^2} \rd y$

is an exact differential equation with solution:

$\map \ln {\dfrac {1 + x y} {1 - x y} } - 2 x = C$


Also see

  • Results about exact differential equations can be found here.


Sources

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