National Institute of Technology Calicut
Department of Mathematics
MA1013E: Mathematics II
Winter Semester 2023-2024: Tutorial sheet 3
1. Using definition, find the Laplace transform of the following functions.
a) sin(3t + 2) b) cosh2 at c) t cos t d) t2 et e) sin2 (at) f) eat cos bt.
2. If L{f (t)} = F (s), then show that
a) L{cosh at f (t)} = 21 [F (s − a) + F (s + a)]
b) L{sinh at f (t)} = 21 [F (s − a) − F (s + a)]
3. If f is a piecewise continuous function on [0, ∞) of exponential order and F (s) =
L{f (t)}, then show that lims→∞ F (s) = 0.
2
4. Check whether the following functions are of exponential order or not? a) tn , n ∈ N b) et .
√
5. Using the fact Γ( 21 ) = π find the Laplace transform of the following functions.
a) f (t) = t−1/2 b) f (t) = t1/2 c) f (t) = 2t1/2 + 8t5/2
6. Find the Laplace transform of the following functions:
Rt Rt
a) e4t sin 2t cos t b) e−2t sin3 t c) 0 eu sinu u du d) e−4t 0 sin 3u
u du
2
e) t3 cos t f) te−2t sin t g) sin 3tt cos t h) sint
√ √
π −1
√ 1
− 4s
. Then show that L( cos√t t ) =
pπ
7. Given that L{sin t} = 34s e se .
2s 2
Rt
8. Find L{sinh ct 0 eau sinh bu du}.
9. Find the convolution of the following functions.
a) 1 ? e−2t b) t ? eat c) eat ? ebt , (a 6= b) d) sin at ? sin at.
10. Find the inverse Laplace transform of the following functions.
s+3 s2 +2s+5 s−a 1
a) (s−1)(s+2) b) (s−1)(s−2)(s−3) c) s2 (s2 +a2 )
d) s2 +6s+15
(s+1)2 6s s−1 s2 +1
e) (s−2)4
f) (s2 −16)2
g) ln s+1 h) ln s(s+1) .
11. Use the convolution theorem to find inverse of Laplace transform of the following
functions
1 s2 1 1
a) s2 s2 +1
b) (s2 +a2 )2
c) s3 s2 +1
d) s2 (s2 −a2 )
12. Find the Laplace transform of the periodic function
(
t if 0 < t < π
f (t) = period 2π
0 if π < t < 2π
1
13. Express the following functions in terms of Heaviside’s unit step function and hence
find the Laplace
( transform. (
8 if t < 2 t − 1 if 1< t < 2
a) f (t) = b) f (t) =
6 if t ≥ 2 3 − t if 2 < t < 3
14. Obtain the solution of the following differential equations by using Laplace transforms.
000 00 0 0 00
a) y − 3y + 3y − y = t2 et , y(0) = 1, y (0) = 0, y (0) = −2.
00 0
b) y + 9y = sin 3t, y(0) = 0, y (0) = 0.
00 0 0
c) y + ty − 2y = 6 − t y(0) = 0, y (0) = 1.
00 0 0
d) ty + 2ty + 2y = 2, y(0) = 1, y (0) is arbitrary.
00 0 0
e) y + 8y + 16y = te−4t y(0) = 1, y (0) = 2.
15. Use the Laplace transform to find the current i(t) in the RLC series circuit when
L = 1h, R = 20Ω, C = 0.005f , E(t) = 150V , t > 0, q(0) = 0, and i(0) = 0.
