Definition:Homogeneous Differential Equation

This page is about Homogeneous Differential Equation. For other uses, see Homogeneous.

Definition

A homogeneous differential equation is a first order ordinary differential equation of the form:

$\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

where both $M$ and $N$ are homogeneous functions of the same degree.

Also presented as

A homogeneous differential equation can also be presented as:

$\dfrac {\d y} {\d x} + \dfrac {\map M {x, y} } {\map N {x, y} } = 0$

or:

$\dfrac {\d y} {\d x} = \dfrac {\map M {x, y} } {\map N {x, y} }$

where both $M$ and $N$ are homogeneous functions of the same degree.

However, note that in the latter case the sign has changed, therefore care needs to be taken when applying the formula.

Some sources present this using the language of differentials in the form:

$\map M {x, y} \rd x = \map N {x, y} \rd y$

or:

$\map M {x, y} \rd x + \map N {x, y} \rd y = 0$

Some sources jump straight to the form:

$\dfrac {\d y} {\d x} = \map F {\dfrac y x}$

Examples

$\paren {x + y} \d x = \paren {x - y} \d y$

is a homogeneous differential equation with general solution:

$\arctan \dfrac y x = \ln \sqrt {x^2 + y^2} + C$

$\paren {x^2 - 2 y^2} \d x + x y \rd y = 0$

is a homogeneous differential equation with solution:

$y^2 = x^2 + C x^4$

$x^2 y' - 3 x y - 2 y^2 = 0$

is a homogeneous differential equation with solution:

$y = C x^2 \paren {x + y}$

$x^2 y' = 3 \paren {x^2 + y^2} \arctan \dfrac y x + x y$

is a homogeneous differential equation with solution:

$y = x \tan C x^3$

$x \sin \dfrac y x \dfrac {\d y} {\d x} = y \sin \dfrac y x + x$

is a homogeneous differential equation with solution:

$\cos \dfrac y x + \ln C x = 0$

$x y' = y + 2 x e^{-y/x}$

is a homogeneous differential equation with solution:

$e^{y / x} = \ln x^2 + C$

Also see

• Solution to Homogeneous Differential Equation

• Results about homogeneous 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: $18.5$: Homogeneous equation
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.7$: Homogeneous Equations
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(3)$ Homogeneous equations
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential equation: differential equations of the first order and first degree: $(3)$ Homogeneous equations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): homogeneous first-order differential equation