Differential Equations Workbook For Dummies®

2.1. The Ins and Outs of Working with Separable Differential Equations

NOTE

If you can separate a differential equation, you're that much closer to solving it. Here's the general form of a separable first order differential equation:

Note that both M(x) and N(y) don't have to be linear in x and y, respectively. For example, you may encounter this differential equation, which is separable but not linear:

You can separate this equation like so:

x dx + y² dy = 0

which gives you

x dx = −y² dy

As you can see, the resulting equation is clearly separated.

Now consider this differential equation:

where

y(0) = 0

You can separate this one into

dy = x² dx

Integrating both sides gives you the following:

When you apply the initial condition, y(0) = 0, you get

c = 0

So the answer is

Here's another example of a typical separable first order differential equation, followed by some practice problems you can work out for yourself.

NOTE

EXAMPLE

Q. Solve this differential equation:

where ...