Section 20.5

Reduction of Order

If you know one solution , you can find the second by assuming .

Introduction

Sometimes we are given one solution to a differential equation (often by inspection or luck), but we need the second linearly independent solution to form the general solution.

Reduction of Order is a technique to find by reducing the problem to a First Order DE.

The Substitution

Recipe

Guess: Let .
Differentiate: Find and using the product rule.
Plug In: Substitute into the original DE.
Reduce: The terms will cancel out, leaving only and terms.
Solve: Let . Solve the linear DE for , then integrate to get .

The Formula

If you carefully carry out the steps for , you get a direct formula for the second solution:

While memorizing the formula is faster, understanding the substitution method is often safer and less prone to calculation errors.

Worked Examples

Example 1: The Origin of "t"

Use reduction of order to find for given .

1. Substitute:

2. Plug in:

Divide by .

3. Simplify:

4. Solve:

Pick simple constants () since we just need one new solution.

. So .

Example 2: Non-Constant Coefficients

Find for given . (for ).

1. Standard Form:

. So .

2. Use Formula:

3. Integrating Factor:

4. Integral:

5. Final:

Ignore constant . .

Example 3: Another Variable Coefficient

Find for given .

1. Substitute:

2. Plug in:

Cancel .

3. Solve for :

4. Integrate for :

. By parts: .

So .

Simplify: use .