23.4 Homogeneous Equations
Sometimes equations can be made separable by changing the variables appropriately. This occurs in the
case of the so called homogeneous equations, those of the form
When this sort of equation occurs, there is an easy trick which will allow you to consider a separable
You define a new variable,
Thus y = ux and so
The variables have now been separated and you go to work on it in the usual way. This method is due to
and dates from around 1691.
Example 23.4.1 Find the solutions of the equation
First note this is of the form
Let u =
and so, separating the variables yields
where K = −C. Hence
Procedure 23.4.2 To solve a homogeneous equation, one which can be placed in the form
do the following:
- Define a new variable v = y∕x. Then y = xv and so y′ = v + xv′.
- Plug in to the equation.
This is separable. Place ∫
before each side and do what it says. Then choose the constant of
integration to satisfy any initial condition which may be present.