* 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*):

This is a *formal* manipulation, since I'm temporarily
treating as a quotient of by . (See the remark below.)

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*.

* Remark.* Here's a justification for the formal
manipulation with and . Think of x and y as depending on a third variables
t, so and . By the Chain Rule,

The initial equation becomes

Then integrate both sides with respect to t.

Then continue as above. In the example that follows, I'll just work formally with and .

* 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

I'll use separation of variables to solve the equations for * exponential growth* and * Neton's
law of cooling*.

Copyright 2018 by Bruce Ikenaga