Homogeneous differential equation

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A differential equation can be homogeneous in either of two respects: the coefficients of the differential terms in the first order case could be homogeneous functions of the variables, or for the linear case of any order there could be no constant term.

Homogeneous type of first-order differential equations

A first-order ordinary differential equation in the form:

${\displaystyle M(x,y)\,dx+N(x,y)\,dy=0}$

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  ${\displaystyle \lambda }$, we find

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

Thus,

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

Solution method

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

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

Introduce the change of variables ${\displaystyle y=ux}$; differentiate using the product rule:

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

thus transforming the original differential equation into the separable form

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

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

The equations in this discussion are not to be used as formulary for solutions; they are shown just to demonstrate the method of solution.

Special case

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

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

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

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

Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If  ${\displaystyle \phi (x)}$  is a solution, so is  ${\displaystyle c\phi (x)}$, where ${\displaystyle c}$ 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 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

${\displaystyle L(y)=0\,}$

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  ${\displaystyle f_{i}}$  of x:

${\displaystyle L=\sum _{i=0}^{n}f_{i}(x){\frac {d^{i}}{dx^{i}}}\,,}$

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

For example, the following differential equation is homogeneous:

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

whereas the following two are inhomogeneous:

${\displaystyle 2x^{2}{\frac {d^{2}y}{dx^{2}}}+4x{\frac {dy}{dx}}+y=\cos(x)\,;}$
${\displaystyle 2x^{2}{\frac {d^{2}y}{dx^{2}}}-3x{\frac {dy}{dx}}+y=2\,.}$

It should be noted that the existence of a constant term is a sufficient condition for an equation to be inhomogeneous, as in the above example.

Notes

1. ^ 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.)