Homogeneous differential equation

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The term "'homogeneous'" is used in more than one context in mathematics. Perhaps the most prominent are the following three distinct cases:

Homogeneous functions
Homogeneous type of first order differential equations
Homogeneous differential equations (in contrast to "inhomogeneous" differential equations). This definition is used to define a property of certain linear differential equations—it is unrelated to the above two cases.

Each one of these cases will be briefly explained as follows.

Homogeneous functions

Main article: Homogeneous function

Definition. A function $f(x)$ is said to be homogeneous of degree $n$ if, by introducing a constant parameter $\lambda$ , replacing the variable $x$ with $\lambda x$ we find:

$f(\lambda x)=\lambda ^{n}f(x)\,.$

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables $f(x,y)$ is said to be homogeneous of degree $n$ if we replace both variables $x$ and $y$ by $\lambda x$ and $\lambda y$ , we find:

$f(\lambda x,\lambda y)=\lambda ^{n}f(x,y)\,.$

Example. The function $f(x,y)=(2x^{2}-3y^{2}+4xy)$ is a homogeneous function of degree 2 because:

$f(\lambda x,\lambda y)=[2(\lambda x)^{2}-3(\lambda y)^{2}+4(\lambda x\lambda y)]=(2\lambda ^{2}x^{2}-3\lambda ^{2}y^{2}+4\lambda ^{2}xy)=\lambda ^{2}(2x^{2}-3y^{2}+4xy)=\lambda ^{2}f(x,y).$

This definition of homogeneous functions has been used to classify certain types of first order differential equations.

Homogeneous type of first-order differential equations

Differential equations
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v t e

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.^[1] That is, multiplying each variable by a parameter $\lambda$ , we find:

$M(\lambda x,\lambda y)=\lambda ^{n}M(x,y)\,.$ and $N(\lambda x,\lambda y)=\lambda ^{n}N(x,y)\,.$

Thus,

${\frac {M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}}={\frac {M(x,y)}{N(x,y)}}\,.$

Solution method

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

${\frac {M(x,y)}{N(x,y)}}={\frac {M(tx,ty)}{N(tx,ty)}}={\frac {M(1,y/x)}{N(1,y/x)}}=f(y/x)\,.$

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

${\frac {d(ux)}{dx}}=x{\frac {du}{dx}}+u{\frac {dx}{dx}}=x{\frac {du}{dx}}+u,$

thus transforming the original differential equation into the separable form:

$x{\frac {du}{dx}}=f(u)-u\,;$

this form can now be integrated directly (see ordinary differential equation).

Special case

A first order differential equation of the form (a, b, c, e, f, g are all constants):

$(ax+by+c)dx+(ex+fy+g)dy=0\,,$

can be transformed into a homogeneous type by a linear transformation of both variables ( $\alpha$ and $\beta$ are constants):

$t=x+\alpha ;\,\,\,\,z=y+\beta \,.$

Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi (x)$ is a solution, so is $c\phi (x)$ , where $c$ is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y; a constant term breaks homogeneity. 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 of the form:

$L(y)=0\,$

where L is a differential operator, a sum of derivatives, each multiplied by a function $f_{i}$ of x:

$L=\sum _{{i=1}}^{n}f_{i}(x){\frac {d^{i}}{dx^{i}}}\,;$

where $f_{i}$ may be constants, but not all $f_{i}$ may be zero.

For example, the following differential equation is homogeneous

$\sin(x){\frac {d^{2}y}{dx^{2}}}+4{\frac {dy}{dx}}+y=0\,,$

whereas the following two are inhomogeneous:

$2x^{2}{\frac {d^{2}y}{dx^{2}}}+4x{\frac {dy}{dx}}+y=\cos(x)\,;$

$2x^{2}{\frac {d^{2}y}{dx^{2}}}-3x{\frac {dy}{dx}}+y=2\,.$

Notes

↑ 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 . (This is a classic reference on ODEs, first published in 1926.)

External links

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