We define what it means for a first order equation to be separable, and we work out solutions to a few examples of separable equations.

Separable Equations

A first order differential equation is separable if it can be written as

where the left side is a product of and a function of and the right side is a function of . Rewriting a separable differential equation in this form is called separation of variables.

To see how to solve (eq:2.2.1), let’s first assume that is a solution. Let and be antiderivatives of and ; that is,

Then, from the chain rule, Therefore (eq:2.2.1) is equivalent to Integrating both sides of this equation and combining the constants of integration yields Although we derived this equation on the assumption that is a solution of (eq:2.2.1), we can now view it differently: Any differentiable function that satisfies (eq:2.2.3) for some constant is a solution of (eq:2.2.1). To see this, we differentiate both sides of (eq:2.2.3), using the chain rule on the left, to obtain which is equivalent to because of (eq:2.2.2).

In conclusion, to solve (eq:2.2.1) it suffices to find functions and that satisfy (eq:2.2.2). Then any differentiable function that satisfies (eq:2.2.3) is a solution of (eq:2.2.1).

Implicit Solutions of Separable Equations

In Examples example:2.2.1 and example:2.2.2 we were able to solve the equation to obtain explicit formulas for solutions of the given separable differential equations. As we’ll see in the next example, this isn’t always possible. In this situation we must broaden our definition of a solution of a separable equation. The next theorem provides the basis for this modification. We omit the proof, which requires a result from advanced calculus called as the implicit function theorem.

It’s convenient to say that (eq:2.2.11) with arbitrary is an implicit solution of . Curves defined by (eq:2.2.11) are integral curves of . If satisfies (eq:2.2.10), we’ll say that (eq:2.2.11) is an implicit solution of the initial value problem (eq:2.2.12). However, keep these points in mind:

  • For some choices of there may not be any differentiable functions that satisfy (eq:2.2.11).
  • The function in (eq:2.2.11) (not (eq:2.2.11) itself) is a solution of .

Constant Solutions of Separable Equations

An equation of the form is separable, since it can be rewritten as However, the division by is not legitimate if for some values of . The next two examples show how to deal with this problem.

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)