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:

  1. Homogeneous functions
  2. Homogeneous type of first order differential equations
  3. 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

Definition. A function    is said to be homogeneous of degree     if, by introducing a constant parameter  , replacing the variable     with     we find:

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables is said to be homogeneous of degree    if we replace both variables    and    by    and  ,  we find:

Example. The function    is a homogeneous function of degree 2 because:


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

Homogeneous type of first-order differential equations

A first-order ordinary differential equation in the form:

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  , we find:

    and    

Thus,

Solution method

In the quotient   , we can let     to simplify this quotient to a function of the single variable :

Introduce the change of variables ; differentiate using the product rule:

thus transforming the original differential equation into the separable form:

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):

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

Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If    is a solution, so is  , where 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:

where L is a differential operator, a sum of derivatives, each multiplied by a function    of x:

where    may be constants, but not all    may be zero.

For example, the following differential equation is homogeneous

whereas the following two are inhomogeneous:

See also

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.)

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