The Characteristic Equation

Guess an exponential and the calculus becomes algebra.

12 problems · about 24 minutes · A-REI.B.4, F-IF.B.6

NameDate

Figure — use these to answer the problems

A real negative root gives pure decay. Make the exponent positive and the solution runs away instead.

y = 1 · -0.75^x + 0

the coefficient1

r, the characteristic root-0.75

the time shown1

Figure 1

Straightforward practice. Get the method working first.

5 problems

y″ + 5y′ + 4y = 0. The roots are −1 and what other value?
Answer
2
What do complex characteristic roots produce?
- Oscillation, with the real part setting growth or decay.
- No solution at all.
y″ + 3y′ + 2y = 0. The roots are −1 and what other value?
Answer
y″ − 4y = 0. The roots are 2 and what other value?
Answer
y″ + y = 0. Are the roots real? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

A repeated root r. What extra factor multiplies the second solution? Enter 1 for t, 2 for t squared.
Answer
y″ + 6y′ + 9y = 0. What is the repeated root?
Answer
Complex roots with real part 0. Does the solution grow, decay or oscillate steadily? Enter 1 grow, 2 decay, 3 steady oscillation.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Match each root situation to the shape of the solution.
Draw a line from each item on the left to its match on the right.
- Two distinct negative real roots
- A repeated root
- Complex roots
- A sum of two decaying exponentials
- An exponential and t times that exponential
- Oscillation, possibly inside an exponential envelope
y″ − 9y = 0. The roots are −3 and what other value?
Answer
The Roots
y″ + 7y′ + 12y = 0. The roots are −3 and what other value?
Answer
The Long Run
Both characteristic roots are negative. What value does the solution approach as t grows large?
Answer