Ordinary Differential Equations
Homogenous Equations Abadi
Universitas Negeri Surabaya
2
A function is called a homogeneous function of the degree n If the following relationship is valid for all :
Example:
is these following function homogenous? If so, in what order?
HOMOGENEOUS FUNCTION
3
Determine whether these following functions are homogenous? If so, in what order?
1.
YOUR TURN
Homogeneous differential equation
A first order differential equation
is called homogeneous equation, if the right side of ODE is a homogeneous function (with respect to the variables and ) of the zero order:
A homogeneous differential equation can be also written in the form or
or alternatively, in the differential form
where are homogeneous functions of the same degree
Solving Homogeneous Differential Equations
A homogeneous equation can be solved by substitute Example :
Solve the differential equation Solution :
It is easy to see that the polynomials and , respectively, at and , are homogeneous functions of the first order.
Therefore, the original differential equation is also homogeneous.
Suppose that , where is a new function depending on . Then
Substituting this into the differential equation, we obtain
Hence , ,
then we have separable differential equation.
Then the solution will be
where is a constant of integration.
Exercises
Solve these ODE problems :
