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
:
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 '' ...
s of the same degree of and .
In this case, the change of variable leads to an equation of the form
:
which is easy to solve by
integration
Integration may refer to:
Biology
* Multisensory integration
* Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technolo ...
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 equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ ...
s, 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 contras ...
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
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...
in section 9 of his 1726 article ''De integraionibus aequationum differentialium'' (On the integration of diff