MATH198 - SOLUTION SHEET 5
1. For each of the functions f (x) specified below, find, showing the details of your work, its Laplace transform F (s). Please use the method specified for each case. (a) f (t) = 2t + 6 Use linearity of the Laplace transform and the Table of Laplace transforms. From the Table L[t] =
1 s2
and L[1] =
1 s
L[2t + 6] =
so 2 6 + 2 s s
(b) f (t) = sin πt Use the Table of Laplace transforms. From the Table L[sin ωt] =
s2
ω so + ω2 L[sin πt] =
s2
π + π2
(c) f (t) = ea−bt
(a, b = const)
Use linearity and the Table. f (t) = ea−bt = ea e−bt a
L[ea−bt ] =
e s+b
so
(d) (
f (t) =
k, if 0 ≤ t < c, (c, k = const) 0, if t > c
Use the definition of the Laplace transform, +∞ Z
f (t)e−st dt.
L[f ] =
0
This one isn’t in our table, so we have to do the integral. L[f ] =
Z ∞
f (t)e
−st
dt =
Z c
0
"
−st
ke
0
k dt = − e−st s
#c
= 0
k 1 − e−sc s
2. Use partial fractions and the Laplace table to find the inverse Laplace transforms of (a) 5 s(s + 2) 5 5 5 = − s(s + 2) 2s 2(s + 2) From the table f (t) =
5 5 − e−2t 2 2
(b) 10 (s + 1)(s2 + 4) You have to try something of the form
A Bs + C + 2 . The result is s+1 s +4
10 2 −2s + 2 = + 2 2 (s + 1)(s + 4) s+1 s +4 From the table, g(t) = 2e−t − 2 cos(2t) + sin(2t)