In the following section we want to discuss a method to solve a certain family of ordinary differential equations – so called
"Separable Differential Equations
"Why separable? Well, with this method our goal is to bring
Often - not always! - a first-order D. E. can be reduced to a form
(I)
Let’s have a practical example: The function
e. g.
can be written as
(just multiply both sides with "y")
Now we easily can identify g(y) and f(y) of equation (I)
® 
; 
Lets go back to the general case:
We can re-write (I) as
(IIa)
or even as
(IIb)
In our example equation, this will result in: 
Integration of both sides of (IIb),
We integrate both sides of the equation:
The left side depends on "y" only, therefore we integrate in respect to y –
The right side depends on "x" only and we integrate it in respect to x.
Did you notice that we added an integration constant on the right hand side only? We actually can do this since it doesn’t really matter if we add a constant of unknown value on each side of the equation or the sum of both ("c") just on the right hand side.
However: Never forget the integration constants! They play a crucial role here!
gives us a new equation
with 
and 
As a result we now have found a new relation between x and y that is free of any derivatives of y.:
in which
Now we are able to solve the new equation algebraically for y in order to obtain a general solution of our D.E.!
Let’s do this with our example!
We had
Now we integrate both sides:
That easy !
Now we can proceed and bring this into a nicer form (Note how important that little integration constant "c" becomes in our calculation! )
are the equations for a family of ellipses.
Let’s see what happens in case we forget the integration constant c:
Then, we would have ended up with
This equation has only one real solution for x=y=0 and describes the origin (0,0). Not exactly a solution you might impress your customers with, not to talk about your boss!
Copyrights 1999, 2000 by Peter Dragovitsch and Ben A. Fusaro