Repeated Roots
What if the quadratic formula gives only one answer? We need a second linearly independent solution, and it comes solely from multiplying by $t$.
Introduction
Consider the case where the discriminant is zero ().
The characteristic equation yields only one root:.
This gives us one solution . But to solve a second order equation, we need two linearly independent solutions.
The Second Solution
Reduction of Order Preview
It turns out (and we will prove this in the next section) that the second solution is simply times the first.
Warning: When finding , don't forget the Product Rule on the second term!
Visual: The "Hump"
Interactive: t dominates initially
For negative roots, basic exponential decay just goes down. Repeated roots can go UP first (due to the term) before eventually decaying.
Worked Examples
Example 1: Solving
Solve .
1. Roots:
.
Double root .
2. Solution:
.
Example 2: Initial Value Problem
Solve , .
1. Roots:
.
.
2. Derivative:
Product rule for second part!
.
.
3. Apply ICs:
.
.
.
4. Final:
.
Example 3: Boundary Values
Solve given .
1. Roots:
.
.
.
2. Apply BCs:
.
So .
.
.
3. Final:
.
Practice Quiz
Loading...