110418 Differential Equations

1.  Solve the system

y_1 ' = 2 y_1 + y_2

y_2 ' = 2 y_2

2.  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?

3.  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.

4.  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.

5.  Repeat the above if f(t) is a sine wave with period pi / 80, and compare.