Definition:Exact Differential Equation/Definition 2

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 known as

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

Also see

• Equivalence of Definitions of Exact Differential Equation

• Results about exact differential equations can be found here.

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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(1)$ Exact equations
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(1)$ Exact equations