Differential Equations Workbook For Dummies®

2.2. Finding Implicit Solutions

Separating differential equations into x and y parts is fine; it can also be quite helpful. Yet sometimes you just can't come up with a neat y = f(x) solution, no matter how hard you try. For example, what if you encounter a differential equation like this one?

You can multiply both sides by dx to get

(1 − y²) dy = x² dx

As you can see, this is a separable differential equation. Integrating both sides gives you

Doesn't exactly look easy to write in the form y = f(x), does it? That's because it's an implicit solution, also known as any solution you can't write like y = f(x). (Solutions that can be written the easy way are considered explicit solutions.)

Although finding implicit solutions can be useful, sometimes you end up having to resort to numerical methods on a computer to convert them into the standard y = f(x) form. On the other hand, finding an implicit solution is occasionally the very best you can do.

Check out how to solve the following implicit solution problem and then try your hand at a few that are just like it.

NOTE

EXAMPLE

Q. Find an implict solution to this differential equation:

A.

1. Multiply both sides by dx:

   (y − y²) dy −