Section 19.2

Separable Differential Equations

If we can get all the y's on one side and all the x's on the other, we can solve by simple integration.

1

Introduction

A Separable Equation is one that can be written in the form:

Or more casually: .

2

The Method

Recipe

  1. Separate: Get all terms involving (including ) on the left, and all terms on the right.
  2. Integrate: Perform .
  3. Solve: You will get an implicit equation . Try to solve for explicitly if possible.

Warning: Don't forget the integration constant immediately after integrating!

3

Visual: Implicit Solutions

Interactive: Contour Map

Often separable equations give us shapes (circles, ellipses) defined implicitly rather than functions y=f(x).

4

Worked Examples

Example 1: Explicit Solution

Solve , .

1. Separate:

.

.

2. Integrate:

.

.

3. Initial Condition:

.

.

4. Solve for y:

.

.

Example 2: Implicit Solution

Solve , .

1. Separate:

.

2. Integrate:

.

3. Initial Condition:

.

.

.

.

4. Final Answer:

.

(Leave in implicit form unless requested otherwise).

Example 3: Factoring Required

Solve .

1. Separate (Algebra first):

.

.

2. Integrate:

.

3. Explicit Solve:

.

5

Practice Quiz

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