Differential Equations Workbook For Dummies®

2.3. Getting Tricky: Separating the Seemingly Inseparable

Sometimes you can convert differential equations that don't look separable into separable ones by using a cool trick. Why would you want to take the time? Because separable equations are usually much easier to solve than differential equations that don't appear separable.

To work some conversion magic on a differential equation, simply substitute y = vx into the equation. Often the result is an easier-to-solve separable equation.

Using y = vx is a useful trick when your differential equation is of the following form:

NOTE

Note that this trick only has a hope of working if f(x, y) = f(tx, ty) where t is a constant (meaning when you put in tx for x and ty for y, the t drops out).

Take a look at this problem:

This differential equation may seem hopelessly inseparable to the uneducated, but lucky for you, you're armed with the y = vx substitution trick!

First things first though: Make sure that f(x, y) = f(tx, ty). Substituting tx for x and ty for y gives you

The t drops out, leaving you with

So f(x, y) = f(tx, ty), ...