Math 2300 Section 005 – Calculus II – Fall 2010
Notes on Separation of Variables – October 27, 2010
Definition: We call a differential equation separable if it can be written in the form
for a function of and a function of .
The Formal Approach to Solving Separable Differntial Equations
If then we can write
Integrating both sides with respect to the variable gives
What will usually happen is that this introduces a natural log in terms of . Exponentiating and solving for then provides a solution in terms of .
Examples:

Using separation of variables, show that solutions to
are circles centered at the origin.
Formal Solution:
Less Formal Solution:

Find the general form of solutions to
where is some constant.
Solution:
Here, satisfies and is simply used to rewrite as either positive or negative.

For find and graph solutions of
The graph of solutions for is as follows.
Of course, as becomes closer to , we see that the slope becomes closer to zero. A general solution is gound by dividing by .
where .