* Separation of variables* is a method for
solving a * differential equation*. I'll
illustrate with some examples.

* Example.* Solve .

"Solve" usually means to find y in terms of x. In general, I'll be satisfied if I can eliminate the derivative by integration.

First, I rearrange the equation to get the x's on one side and the
y's on the other (*separation*):

Next, I *integrate* both sides:

I only need an arbitrary constant on one side of the equation.
Finally, I *solve* for y in terms of x, if possible:

Here's a convenient trick which I'll use in these situations. Think of as . Move the to the other side:

Now *define* :

The last step makes the equation nicer, and it's easier to solve for
the arbitrary constant when you have an *initial value
problem*.

* Example.* Solve , where .

Separate:

Integrate:

In this case, solving would produce plus and minus square roots, so I'll leave the equation as is.

Plug in the initial condition: When , :

Hence, the solution is

Copyright 2005 by Bruce Ikenaga