Question:
`q001. Solve the system
y_1 ' = 2 y_1 + y_2
y_2 ' = 2 y_2
Your solution:
Confidence rating:
Given Solution:
Self-critique (if necessary):
Self-critique rating:
Question:
`q002. Translate the above system into a matrix equation and solve by finding eigenvalues and eigenvectors. You will encounter a problem with this method; you might have to attempt the solution to recognize the problem. What is the problem?
Your solution:
Confidence rating:
Given Solution:
Self-critique (if necessary):
Self-critique rating:
Question:
`q003. In the preceding you obtained one solution to the eigenvalue problem. You might well have found a second solution as well. However the Wronskian of your fundamental set would have been zero, indicating that your solutions were not linearly independent.
See if you can find a second solution of the form y_2 = t y_1 + y_1, where y_1 is your first solution.
Your solution:
Confidence rating:
Given Solution:
Self-critique (if necessary):
Self-critique rating:
Question:
`q004. If f(t) is periodic with period T, then as anticipated in the preceding class the Laplace Transform of each period is changes by factor e^-(s T). The reason for this is a straightforward result of the integral that defines the transform.
So with successive periods the transform is changed by factors e^(-s T), e^(-s * 2 T), e^(-s * 3 T), etc..
The transform over the first period is just the transform integral from 0 to T: integral( f(t) e^(-s t) dt, t from 0 to T).
So the transform of the periodic function is
integral( f(t) e^(-s t) dt, t from 0 to T) * (1 + e^(-s T) + e^(-2 s T) + e^(-3 s T) + ... )
= integral( f(t) e^(-s t) dt, t from 0 to T) * (1 + e^(-s T) + (e^(-s T))^2 + (e^(-s T))^3 + ... ).
1 + e^(-s t) + (e^(-s t))^2 + (e^(-s t))^3 + ... is a geometric series with common ratio e^(-s T), and is therefore equal to 1 / (1 - e^(-s T)).
It follows that
L(f(t)) = integral( f(t) e^(-s t) dt, t from 0 to T) / (1 - e^(-s T)).
Use this fact to find the solution to the equation
y '' + omega * y = f(t),
where omega = 100 rad/s and f(t) is an alternating square wave with amplitude 3 and period pi/80.
Then find the solution for
y '' + delta y ' + omega y = f(t)
for the same value of omega and the same function f(t), with delta taking the half value necessary to critically damp the system.
Repeat of delta is 10% greater than the value necessary for critical damping.
Repeat once more if delta is equal to the value necessary for critical damping.
Intepret your solutions.
Your solution:
Confidence rating:
Given Solution:
Self-critique (if necessary):
Self-critique rating:
Question:
`q005. Repeat the above if f(t) is a sine wave with period pi / 80, and compare.
Your solution:
Confidence rating:
Given Solution:
Self-critique (if necessary):
Self-critique rating: