Definition:Exact Differential Equation/Also presented as

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 $\dfrac {\partial f} {\partial y}$.

That is, such that:

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

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 18$: Basic Differential Equations and Solutions: $18.4$: Exact equation
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.8$: Exact Equations