Definition:Homogeneous Differential Equation

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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:

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


Also see

  • Results about homogeneous differential equations can be found here.


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:

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


Sources

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