Section 20.2
Real Distinct Roots
Turning calculus into algebra. How a quadratic equation unlocks the solution to constant coefficient DEs.
1
Introduction
We are looking at linear, homogeneous equations with constant coefficients:
We guess a solution of the form . Plugging this in gives:
Since , we get the Characteristic Equation:.
2
The Method
Recipe
- Write Characteristic Eq: Replace with , with , and with .
- Find Roots: Solve for using factoring or the quadratic formula.
- Case 1: Two Distinct Real Roots ($b^2 - 4ac > 0$)
The general solution is:
3
Visual: Exponential Behavior
Interactive: Changing Roots
Negative roots (e.g., friction) lead to decay. Positive roots lead to unbounded growth.
4
Worked Examples
Example 1: Basic Roots
Solve .
1. Characteristic Equation:
.
2. Factor:
.
Roots: , .
3. Solution:
.
Example 2: Initial Value Problem
Solve with .
1. Roots:
.
.
2. Apply ICs:
.
.
.
3. Solve System:
and .
Add them: .
Sub in: .
4. Final:
.
Example 3: Boundary Value Problem
Solve with .
1. Roots:
.
.
2. Apply BCs:
.
.
3. Deduce:
By inspection, if , second eq is which matches.
First eq matches.
So .
5
Practice Quiz
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