Definition:Exact Differential Equation
Definition
Definition 1
An exact differential equation is a first order ordinary differential equation in which the total differential is equal to zero:
- $\dfrac {\partial f} {\partial x} \rd x + \dfrac {\partial f} {\partial y} \rd y = 0$
Definition 2
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 often referred to just as 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.