Task 1
Determine the Laplace transform of the functions: a)
b) d)
(t) = 4t + tooaa + —. 2
h(t) cos + 4) = 3 + + t sin(1t). fa(t) =
e) Rewrite the function using the Heaviside’s function and determine the Laplace transform of the function:
3t-1 1<t<2 h(t)= tel—t 2 < t < 3 . 1 ellers

Task 2 Determine the inverse Laplace transform of the functions: a)
Fibs) 12a — 24 .„ _ ” „
b)
F2(,) 54+ When using s-shift determine the inverse Laplace transform of the functions:
0)
Ss — 2

F2(s) + + When using s-shift determine the inverse Laplace transform of the functions:
4) Task 3
38 — 2 F.3(0) (2s +
2s + 1 F4(s)20 + 24s + 97
a) Determine the convolution of the function using the Laplace transformation. `Mu will need the result from problem 2b) f. * b) Solve for x (t) using the Laplace transform. Hint: identify the convolution in the equation. X(i) = t — 3 j'(t — r)x(r)d,r. .

Task 4 Use the Laplace transform to solve the differential equation y”(t) — y'(t) — 2y(t) = e’tt(t — 2) + 13(t — 4),
with the initial values
v(0) = 0,