DEPARTMENT OF MATHEMATICS

School of Advanced Sciences

Applications of Differential and Difference Equations
Course Code - MAT 1002
Winter Semester 2024 - 25

Problem sheet 2 - Laplace Transforms (Module - 2)

Course Intructor: Dr. Ramesh Arumugam Slot: A1/A2

1. Find the Laplace transform of each of the following functions

(a) e3t t4 (b) e−2t cos 4t (c) e2t (t − 1)2 (d) t2 sin t (e) te−t sin 2t
t
cos(2t) − cos (3t)
Z
a
(f ) t , a > 0 (g) cosh(at) sin bt (h) (i) cos 2u du
t 0
( Z t
sin t, 0 < t < π
(j) f (t) = with f (t) = f (t+2π) (k) (t−u)2 cos u du
−sin t, π < t < 2π 0

5
X
(l) f (t) = u0 (t) + (−1)k uk (t) (m) (t + 3)u7 (t) (n) 3δ(t − π) − δ(t − 2π)
k=1

2. Determine the inverse Laplace transform of each of the following functions

2s + 1 s−1 9+s 1 + e−2s

(a) (b) (c) (d) − 5 (e)
s(s2 + 9) (s + 1)3 4 − s2 s2 + 16

s2 + s + 1 1 − s2
 
1 s+a 1 1
(f ) 2 (g) (h) (i) log (j) sin
s2 + 3s + 2 (s − 5)n (s2 + 1)2 s+b s s

3. Determine the initial and final values of f (t) if

10(s + 1) 10(s + 2) 2s2

(a) F (s) = (b) F (s) = (c) F (s) =
s(s2 + 2s + 1) (s + 1)(s + 4) (s + 1)(s2 + 2s + 2)
 4. Suppose Laplace transform of a function f (t) is given by F (s) = s(s218+36) , find f (0) and
f 0 (0) using initial value theorem and Laplace transform of derivatives.

5. Using Convolution Theorem, find

s2
     
−1 1 −1 3s −1
(a) L (b) L (c) L
s3 (s2 + 1) (s2 + 1)2 (s2 + 9)(s2 + 25)

6. Express in terms of Heaviside’s unit step function and find the Laplace transform

1 − cos(t), 0 ≤ t < π

f (t) = 2 + sin(2t), π ≤ t < 2π

3 − t, t ≥ 2π


7. Use Laplace transforms to solve the initial-value problem

y 0 + y = 5u(t − 1), y(0) = 4.

8. Use Laplace transforms to solve the initial-value problem

y 0 + y = e−3t cos 2t, y(0) = 0.

9. Solve
d2 y
2
+ 4y = cos 2t, y(0) = −2, y 0 (0) = 0.
dt

10. Find the solution of the initial value problem and sketch the graph of the solution for

y 00 + y = δ(t − 2π) cos t, y(0) = 0, y 0 (0) = 0.

11. Solve
d2 y dy
2
−6 + 9y = t2 e3t , y(0) = 2, y 0 (0) = 6.
dt dts

12. Determine the response of the damped mass-spring system under a square wave, modeled
as
y 00 + 3y 0 + 2y = u(t − 1) − u(t − 2).

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13. Solve
y 00 + ty 0 − 2y = 4, y(0) = −1, y 0 (0) = 0.

14. Consider a spring mass system where m = 1, k = 13, and c = 4. Assume that the mass
is initially displaced 1 m and released. Finally, assume that at t = 3, the mass is struck
with a hammer in the positive direction (i.e., f (t) = δ(t − 3)). Set up and solve an
initial-value problem that describes this situation using Laplace transform.

15. Consider an undamped spring-mass system with spring constant k = 4. Suppose that
the mass is displaced 1 unit from equilibrium and struck with a force to impart an initial
velocity of y 0 (0) = 1. In addition, at times t = 7 and t = 20, a hammer delivers a one-
unit impulse to the mass in the positive direction. Assuming consistent units, solve an
IVP that models this situation (hint: y 00 + 4y = δ(t − 7) + δ(t − 20), y(0) = 1, y 0 (0) = 1).

16. Find the solution of the initial-value problem

∞
d2 y X
+ 2y = δn (t), y(0) = 0, y 0 (0) = 0, where δn (t) = δ(t − n).
dt2 n=1

What is the long-term qualitative behavior of this solution?

17. Solve
y 00 + 4y 0 + 3y = 1 − u(t − 2) − u(t − 4) + u(t − 6), y(0) = 0, y 0 (0) = 0.

18. The differential equation for the instantaneous charge q(t) on the capacitor in an LRC-
series circuit is given by
d2 q dq 1
L 2 + R + q = E(t).
dt dt C
Use the Laplace transform to find q(t) when L = 1 h, R = 20 Ω, C = 0.005 f ,
E(t) = 150 V , q(0) = 0, and i(0) = 0. What is the current i(t)?

19. In an RLC circuit, R = 110 Ω ohms, L = 1 H, C = 0.001 F , and a battery supplying

E0 = 90 V . Suppose that the circuit is initially passive – no current and no charge. At
time t = 0, the switch is closed, and at time t = 1 it is opened and left open. Find the
resulting current i(t) in the circuit.

20. Use the Laplace transform to solve the given integral equation or integrodifferential
equation Z t
f (t) + 2 f (u) cos(t − u)du = 4e−t + sin t.
0

Best wishes

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