Section 19.3

Exact Differential Equations

Reversing the Chain Rule. If a DE looks like , we can solve it by finding .

1

Introduction

Recall from Calculus III (Chapter 16) that if is conservative, then .

An Exact Differential Equation is just where vector field is conservative. The solution is the level curves of the potential function: .

2

Test & Method

The Test

The equation is exact if and only if:

The Solution

  1. Check .
  2. Integrate with respect to : .
  3. Differentiate w.r.t and set equal to .
  4. Solve for , then .
  5. Write solution .
3

Visual: Level Curves

Interactive: Potential Function

4

Worked Examples

Example 1: Basic Exact

Solve .

1. Check Exactness:

.

.

Matches! It's exact.

2. Find Potential :

Integrate M w.r.t x: .

3. Use N to find :

.

Set equal to N: .

.

.

4. Final Solution:

.

Example 2: Initial Value Problem

Solve , .

1. Rewrite Form:

(Careful with signs!)

Wait, RHS was . Bring to left: .

Yes. .

2. Check:

.

. Exact.

3. Integrate:

.

.

.

.

4. IVP:

.

.

.

Example 3: Trigonometric Form

Solve .

1. Check Exactness:

.

.

Match! It is exact.

2. Integrate M:

.

(Be careful: ).

3. Differentiate & Match:

.

.

.

4. Final:

.

5

Practice Quiz

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