110413 Differential Equations

1.  Find solutions of

y ' =  [ 5, 3; -4, -3 ] y

for which y(1) = [2; 0].

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Find the expression for W(t), the Wronskian of the solution.

Show that W(t) = W(1) * e^(integral(2 ds, s from 1 to t)).

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2. Recall that a geometric series is of the form 1 + r + r^2 + r^3 + .... r^n.

The sum of this geometric series is (1 - r^(n+1) ) / (1 - r).

In the limit as n -> infinity, the sum is 1 / (1 - r).

What therefore is the sum of the series

1 - e^(-s) + e^(-2 s) - e^(-3 s)  ... ?

What might this have to do with the Laplace Transform of the step function H(t) - 2 H ( t - 1) + 2 H ( t - 2) ... ?

3.  Here is a table of some functions and their Laplace Transforms, all easily found by straightforward integration:

If the Laplace Transform of a function f(t) is found to be 1 / (s - 4) + 1 / (s + 1), then what is f(t)?

What if the Laplace Transform is 2 / (s - 4) + 3 / (s + 1)?

What if the Laplace Transform is 5 / (s^2 + 25)?

What if the Laplace Transform is 10 / (s^2 + 25)?

What if the Laplace Transform is s / (s^2 + 25)?

4.  What is the Laplace Transform of the equation

y '' - 4 y ' - 5 y = 0

subject to y (0) = 0 and y ' (0) = 1?  Denote the Laplace Transform of y by Y.

Solve for Y, and find the function y corresponding to this transform.

5.  Let f(t) = 0 everywhere except on the interval 0 <= t <= 4; on this interval f(t) = t.  Sketch a graph of f(t).

What is the Laplace Transform of f(t)?

Let g(t) = 0 everywhere except on the interval 0 <= t <= 4, on which g(t) = t, and on the interval 4 <= t <= 8, on which g(t) = t - 4. 

Sketch a graph and find the Laplace Transform of g(t).

What would happen if we kept adding intervals in a similar manner?

6.  Let f(t) = 0 everywhere except on the interval 0 <= t <= 4, on which interval f(t) = cos( pi / 2 * t).  Sketch the graph of this function and find its Laplace Transform.

Let g(t) = 0 everywhere except on the interval 0 <= t <= 8, on which interval g(t) = cos( pi / 2 * t).  Sketch the graph of this function and find its Laplace Transform.

What would happen if we kept adding intervals in a similar manner?

There might be some bad arithmetic in the above; you should correct the solution.

Overview:  We are assuming solutions to the equation A y = y ' to be of the form e^(lambda t) * [ c ; d ], leading to y ' = lambda e^(lambda t) * [ c; d ] so that A y = lambda y.  This leads to the eigenvalue problem; having found an eigenvalue lambda we can then solve for c and d.  We expect two eigenvalues for the system of two equations, so we expect two solutions.  If the solutions are denoted y_1 = [y_11; y_21] and y_2 = [y_12; y_22] then the Wronskian is the matrix [ y_11, y_12; y_21, y_22 ].

If 1 / (s - 4)  + 1 / (s + 1) is the Laplace Transform of a solution, then since L(e^(alpha t)) = 1 / (s - alpha), 1 / (s - 4) is the transform of e^(4 t) and 1 / (s + 1) is the transform of e^-t.

By the linearity of the transform, 1 / (s - 4)  + 1 / (s + 1) is thus the transform of e^(4 t) + e^(-t).

In the above, A and B can be found using partial fractions.