By definition, a homogeneous function of degree satisfies the property
For example, the functions
are all homogeneous functions, of degrees three, two and three respectively (verify this assertion).
Observe that any homogeneous function of degree can be equivalently written as follows:
Having seen homogeneous functions we define homogeneous DEs as follows :
Any DE of the form or is called homogeneous if and are homogeneous functions of the same degree.
What is so special about homogeneous DEs ? Well, it turns out that they are extremely simple to solve. To see how, we express both and as, say and . This can be done since and are both homogeneous functions of degree . Doing this reduces our DE to
The function stands for
Now, the simple substitution reduces this DE to a VS form :
Thus, transforms to
This can now be integrated directly since it is in VS form.
Let us see some examples of solving homogeneous DEs
Solve the DE
This is obviously a homogeneous DE of degree one since the RHS can be written as
Using the substitution reduces this DE to
Using above, we have
Integrating, we have
Substituting for , we finally obtain the required general solution to the DE: