Section 20.6

Fundamental Sets of Solutions

We formalize the requirements for a "Complete" solution using the Wronskian and Abel's Theorem.

Introduction

A pair of solutions forms a Fundamental Set of Solutions if the general solution can satisfy any initial conditions .

This is true if and only if the Wronskian is non-zero:.

Abel's Theorem

Statement

If are solutions to , then the Wronskian is given by:

Implication: Since exponentials are never zero, is either always zero (if ) or never zero (if ).
You only need to check independence at a single point!

Visual: Linear Independence

Interactive: Parallel Functions?

For functions to be linearly dependent, one must be a constant multiple of the other: .

Worked Examples

Example 1: Checking Fundamental Set

Check if and form a fundamental set for .

1. Check if solutions:

: . Yes.

2. Check Wronskian:

3. Conclusion:

The Wronskian is non-zero for . They form a fundamental set of solutions on any interval not containing .

Example 2: Abel's Theorem

Without solving, find the Wronskian of up to a constant.

1. Standard Form:

. So .

2. Apply Theorem:

Example 3: Constructing General Solution

Given and are solutions to , find the solution where .

1. Wronskian Check:

. Non-zero for .

2. Apply ICs:

3. Derivative:

4. Solve: