Definition:Exact Differential Equation

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.

Also see

 * Solution to Exact Differential Equation