Determine the laplace transform you may use the laplace

Question

1. Find the Laplace transform of f(t) using the integral definition of the Laplace transform

                e^-3t,        0 < t < 3

f(t) =

                1,               3 < t

2. Determine the Laplace transform. You may use the Laplace transform table.

L[t² - t + e^-3t cos2t]

3. Solve the IVP (find y(t)) using the method of the Laplace transform.

y'' + y = t2 + 2;                   y(0) = 1; y'(0) = -1

Hint: You will NOT need to use partial fractions

4. Solve for Y(s). You do not need to do a partial fraction expansion, but express your answer as a single rational function with series of factors in the denominator and a number of summed (or differenced) terms in the numerator.

y'' - 6y' + 5y = tcos(2t);                                 y(0) = 2; y'(0) = -1

5. Consider g(t) below.

a. Plot g(t),

b. Express g(t) using step functions

c. Find the Laplace transform

                5 cos(t),                  0 ≤ t ≤ 2π

g(t) =

                t,                             2π < t

6. Find L^-1[s(e^-2s)/s²+9]

7. Solve for y(t).

y'' + y = tu(t-3);                 y(0) = 0, y'(0) = 1

8. Solve for y(t)).

y'' - y = sin(t-3)u(t-3);                    y(0) = 0; y'(0) = 1