# Definition:Exact Differential Equation

## Contents

- 1 Definition
- 2 Also presented as
- 3 Also known as
- 4 Examples
- 4.1 $e^y \rd x + \paren {x e^y + 2 y} \rd y = 0$
- 4.2 $\paren {x + \dfrac 2 y} \rd y + y \rd x = 0$
- 4.3 $\paren {y - x^3} \rd x + \paren {x + y^3} \rd y = 0$
- 4.4 $\paren {y + y \cos x y} \rd x + \paren {x + x \cos x y} \rd y = 0$
- 4.5 $\paren {\sin x \sin y - x e^y} \rd y = \paren {e^y + \cos x \cos y} \rd x$
- 4.6 $-\dfrac 1 y \sin \dfrac x y \rd x + \dfrac x {y^2} \sin \dfrac x y \rd y = 0$
- 4.7 $\paren {2 x y^3 + y \cos x} \rd x + \paren {3 x^2 y^2 + \sin x} \rd y = 0$
- 4.8 $\d x = \dfrac y {1 - x^2 y^2} \rd x + \dfrac x {1 - x^2 y^2} \rd y$

- 5 Also see
- 6 Sources

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $18.4$: Basic Differential Equations and Solutions - 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 2.8$: Exact Equations - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**differential equation**