Homogeneous differential equation

Short description: Type of ordinary differential equation

A differential equation can be homogeneous in either of two respects.

A first order differential equation is said to be homogeneous if it may be written f(x,y)dy=g(x,y)dx, where f and g are homogeneous functions of the same degree of x and y.[1] In this case, the change of variable y = ux leads to an equation of the form dxx=h(u)du, which is easy to solve by integration of the two members.

Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.

## Contents

* 1 History
* 2 Homogeneous first-order differential equations
* 2.1 Solution method
* 2.2 Special case
* 3 Homogeneous linear differential equations
* 4 See also
* 5 Notes
* 6 References
* 7 External links

## History

The term homogeneous was first applied to differential equations by Johann Bernoulli in section 9 of his 1726 article De integraionibus aequationum differentialium (On the integration of differential equations).[2]

## Homogeneous first-order differential equations

Differential equations

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

* Natural sciences
* Engineering

* Astronomy
* Physics
* Chemistry

Biology
* Geology

Applied mathematics

* Continuum mechanics
* Chaos theory
* Dynamical systems

Social sciences

* Economics
* Population dynamics

Classification

Types

* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear

By variable type

* Dependent and independent variables

* Autonomous
* Complex
* Coupled / Decoupled
* Exact
* Homogeneous / Nonhomogeneous

Features

* Order
* Operator

* Notation

Relation to processes

* Difference (discrete analogue)

* Stochastic
* Stochastic partial
* Delay

Solution

General topics

* Picard–Lindelöf theorem
* Wronskian

* Phase portrait
* Phase space

* Lyapunov / Asymptotic / Exponential stability
* Rate of convergence
* Series / Integral solutions

* Numerical integration
* Dirac delta function

Solution methods

* Inspection
* Method of characteristics

Euler
* Exponential response formula
* Finite difference (Crank–Nicolson)
* Finite element
* Infinite element
* Finite volume
* Galerkin
* Petrov–Galerkin
* Integrating factor
* Integral transforms
* Perturbation theory
* Runge–Kutta

* Separation of variables
* Undetermined coefficients
* Variation of parameters

People

* Isaac Newton
* Gottfried Leibniz
* Leonhard Euler
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

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A first-order ordinary differential equation in the form:

M(x,y)dx+N(x,y)dy=0

is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n.[3] That is, multiplying each variable by a parameter λ, we find

M(λx,λy)=λnM(x,y)andN(λx,λy)=λnN(x,y).

Thus, M(λx,λy)N(λx,λy)=M(x,y)N(x,y).

### Solution method

In the quotient M(tx,ty)N(tx,ty)=M(x,y)N(x,y), we can let t = ⁠1/x⁠ to simplify this quotient to a function f of the single variable ⁠y/x⁠:

M(x,y)N(x,y)=M(tx,ty)N(tx,ty)=M(1,y/x)N(1,y/x)=f(y/x). That is dydx=−f(y/x).

Introduce the change of variables y = ux; differentiate using the product rule:

dydx=d(ux)dx=xdudx+udxdx=xdudx+u.

This transforms the original differential equation into the separable form xdudx=−f(u)−u, or 1xdxdu=−1f(u)+u, which can now be integrated directly: ln x equals the antiderivative of the right-hand side (see ordinary differential equation).

### Special case

A first order differential equation of the form (a, b, c, e, f, g are all constants) (ax+by+c)dx+(ex+fy+g)dy=0 where af ≠ be can be transformed into a homogeneous type by a linear transformation of both variables (α and β are constants): t=x+α;z=y+β, where α=cf−bgaf−be;β=ag−ceaf−be. For cases where af = be, introduce the change of variables u = ax \+ by or u = ex \+ fy; differentiation yields: dudx=a−b(ac+auag+eu), or dudx=e−f(ec+aueg+eu), for each respective substitution. Both may be solved via separation of variables.

## Homogeneous linear differential equations

See also: Linear differential equation

A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or any derivative of it. A linear differential equation that fails this condition is called inhomogeneous.

A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is

L(y)=0

where L is a differential operator, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function fi of x:

L=∑i=0nfi(x)didxi, where fi may be constants, but not all fi may be zero.

For example, the following linear differential equation is homogeneous:

sin⁡(x)d2ydx2+4dydx+y=0,

whereas the following two are inhomogeneous:

2x2d2ydx2+4xdydx+y=cos⁡(x);

2x2d2ydx2−3xdydx+y=2. The existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.

## See also

* Separation of variables

## Notes

1. ↑ Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 978-1-285-40110-2. https://books.google.com/books?id=pasKAAAAQBAJ&q=homogeneous.
2. ↑ "De integraionibus aequationum differentialium". Commentarii Academiae Scientiarum Imperialis Petropolitanae 1: 167–184. June 1726. https://archive.org/details/commentariiacade01impe.
3. ↑ Ince 1956, p. 18

## References

* Boyce, William E.; DiPrima, Richard C. (2012), Elementary differential equations and boundary value problems (10th ed.), Wiley, ISBN 978-0470458310 . (This is a good introductory reference on differential equations.)
* Ince, E. L. (1956), Ordinary differential equations, New York: Dover Publications, ISBN 0486603490, https://archive.org/details/ordinarydifferen029666mbp . (This is a classic reference on ODEs, first published in 1926.)
* Andrei D. Polyanin; Valentin F. Zaitsev (15 November 2017). Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. CRC Press. ISBN 978-1-4665-6940-9. https://books.google.com/books?id=L3JQDwAAQBAJ&q=homogeneous.
* Matthew R. Boelkins; Jack L. Goldberg; Merle C. Potter (5 November 2009). Differential Equations with Linear Algebra. Oxford University Press. pp. 274–. ISBN 978-0-19-973666-9. https://books.google.com/books?id=s1-mZMg5fLgC&q=homogeneous&pg=PA274.

## External links

* Homogeneous differential equations at MathWorld
* Wikibooks: Ordinary Differential Equations/Substitution 1

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Differential equations

Classification|

| Operations|

* Differential operator
* Notation for differentiation
* Ordinary
* Partial
* Differential-algebraic
* Integro-differential
* Fractional
* Linear
* Non-linear
* Holonomic

|

Attributes of variables|

* Dependent and independent variables
* Homogeneous
* Nonhomogeneous
* Coupled
* Decoupled
* Order
* Degree
* Autonomous
* Exact differential equation
* Complex differential equation

Relation to processes|

* Difference (discrete analogue)
* Stochastic
* Stochastic partial
* Delay

Solutions|

| Solution topics|

* Picard–Lindelöf theorem (existence and uniqueness)
* Wronskian
* Phase portrait
* Phase space
* Lyapunov stability
* Asymptotic stability
* Exponential stability
* Rate of convergence
* Series solutions
* Integral solutions
* Numerical integration
* Dirac delta function

|

Solution methods|

* Inspection
* Separation of variables
* Method of undetermined coefficients
* Variation of parameters
* Integrating factor
* Integral transforms
* Euler method
* Finite difference method
* Crank–Nicolson method
* Runge–Kutta methods
* Finite element method
* Finite volume method
* Galerkin method
* Perturbation theory

Applications|

* List of named differential equations

Mathematicians|

* Isaac Newton
* Gottfried Wilhelm Leibniz
* Leonhard Euler
* Jacob Bernoulli
* Émile Picard
* Józef Maria Hoene-Wroński
* Ernst Lindelöf
* Rudolf Lipschitz
* Joseph-Louis Lagrange
* Augustin-Louis Cauchy
* John Crank
* Phyllis Nicolson
* Carl David Tolmé Runge
* Martin Kutta

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