Homogeneous differential equation
Encyclopedia
The term homogeneous differential equation has several distinct meanings.
One meaning is that a first-order ordinary differential equation
is homogeneous (of degree 0) if it has the form
where
is a homogeneous function
of degree zero; that is to say, that
.
In a related, but distinct, usage, the term linear homogeneous differential equation is used to describe differential equation
s of the form
where the differential operator
L is a linear operator, and y is the unknown function.
The remainder of this article is about homogeneous differential equations in the first sense defined above.
for all t, so t can be arbitrarily chosen to simplify the form of the equation. One can solve this equation by making a simple change of variables
, and then using the product rule
on the left hand side as follows,
and then using the identity
to simplify the right hand side by choosing to set 
to be 
, transforming the original problem into the separable
differential equation
which can then be integrated by the usual methods.
One meaning is that a first-order ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
is homogeneous (of degree 0) if it has the form
where
is a homogeneous functionHomogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree zero; that is to say, that
.In a related, but distinct, usage, the term linear homogeneous differential equation is used to describe differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s of the form
where the differential operator
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, accepting a function and returning another .This article considers only linear operators,...
L is a linear operator, and y is the unknown function.
The remainder of this article is about homogeneous differential equations in the first sense defined above.
Solving homogeneous differential equations
By the definition above, it can be seen that for all t, so t can be arbitrarily chosen to simplify the form of the equation. One can solve this equation by making a simple change of variablesChange of variables
In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with new ones; the new and old variables being related in some specified way...
, and then using the product ruleProduct rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
on the left hand side as follows,
-
.
and then using the identity
to simplify the right hand side by choosing to set to be , transforming the original problem into the separableSeparation of variables
In mathematics, separation of variables 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 on a different side of the equation....
differential equation
which can then be integrated by the usual methods.