11.1.1 Problem 1

1.a. On page 5 we found that:

Using the dilation property of the Laplace transform, we find that:

1.b. We find that:

1.c. We find that:

1.d. We find that:

11.1.2 Problem 3

3. We must solve the ODE:

subject to the initial conditions y(0) = y ?(0) = 0. Taking the Laplace transform of both sides (and noting that the Laplace transform of 1 and of u( t) must be the same because the functions are identical when t > 0), making use of the initial conditions, and denoting the Laplace transform of y( t) by Y( s), we find that:

Thus, we find that:

After clearing denominators we find that:

Rewriting this, we find that:

Equating coefficients, we find that:

We find that A = 1/4, B = ?1/3, and C = 1/12. Thus:

We see that:

11.1.3 Problem 5

5. We must solve the integral equation:

Taking Laplace transforms and denoting the Laplace transform of y( t) by Y( s), we find that:

Rearranging the terms, we find that:

We find that:

Clearing the denominators, we find that:

Rewriting this, we find that:

Equating coefficients we find that:

Solving these equations we find that:

We find that:

By inspection, the inverse transform of this is just:

11.1.4 Problem 7

7. Following the instructions given in the problem, we find that:

11.2.1