HOME

TheInfoList



OR:

A
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
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 and are
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
s of the same degree of and . In this case, the change of variable leads to an equation of the form :\frac = 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 In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.


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).


Homogeneous first-order differential equations

A first-order
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
in the form: :M(x,y)\,dx + N(x,y)\,dy = 0 is a homogeneous type if both functions and are
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
s of the same degree . That is, multiplying each variable by a parameter , we find :M(\lambda x, \lambda y) = \lambda^n M(x,y) \quad \text \quad N(\lambda x, \lambda y) = \lambda^n N(x,y)\,. Thus, :\frac = \frac\,.


Solution method

In the quotient \frac = \frac, we can let to simplify this quotient to a function of the single variable : :\frac = \frac = \frac=f(y/x)\,. That is :\frac = -f(y/x). Introduce the change of variables ; differentiate using the product rule: :\frac=\frac = x\frac + u\frac = x\frac + u. This transforms the original differential equation into the separable form : x\frac = -f(u) - u, or : \frac 1x\frac = \frac , which can now be integrated directly: equals the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
of the right-hand side (see
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
).


Special case

A first order differential equation of the form (, , , , , are all constants) : \left(ax + by + c\right) dx + \left(ex + fy + g\right) dy = 0 where can be transformed into a homogeneous type by a linear transformation of both variables ( and are constants): :t = x + \alpha; \;\; z = y + \beta \,.


Homogeneous linear differential equations

A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if is a solution, so is , for any (non-zero) constant . 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
acting on where is usually the independent variable and is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is : L(y) = 0 where is a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
, a sum of derivatives (defining the "0th derivative" as the original, non-differentiated function), each multiplied by a function of : : L = \sum_^n f_i(x)\frac \, , where may be constants, but not all may be zero. For example, the following linear differential equation is homogeneous: : \sin(x) \frac + 4 \frac + y = 0 \,, whereas the following two are inhomogeneous: : 2 x^2 \frac + 4 x \frac + y = \cos(x) \,; : 2 x^2 \frac - 3 x \frac + 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 In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...


Notes


References

* . (This is a good introductory reference on differential equations.) * . (This is a classic reference on ODEs, first published in 1926.) * *


External links


Homogeneous differential equations at MathWorldWikibooks: Ordinary Differential Equations/Substitution 1
{{Differential equations topics Differential equations Ordinary differential equations