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Problem Sheet 2 ME A

The document is a problem sheet for a Complex Analysis and Differential Equations course at the National Institute of Technology Tiruchirappalli. It contains various problems related to finding Laplace transforms, inverse Laplace transforms, and solving differential equations using the Laplace transform method. The problems are structured for students in the I B.Tech ME-A Semester-II January 2025 session.

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keshavsudharsan06
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20 views2 pages

Problem Sheet 2 ME A

The document is a problem sheet for a Complex Analysis and Differential Equations course at the National Institute of Technology Tiruchirappalli. It contains various problems related to finding Laplace transforms, inverse Laplace transforms, and solving differential equations using the Laplace transform method. The problems are structured for students in the I B.Tech ME-A Semester-II January 2025 session.

Uploaded by

keshavsudharsan06
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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National Institute of Technology Tiruchirappalli

Department of Mathematics
Complex Analysis and Differential Equations (MAIR21)
I B.Tech ME-A Semester-II January 2025 Session
Problem Sheet-2

Find the Laplace transform of the following functions:


√ cos
√
1) t sin at 2) t cos at 3) ( t − 1 3
√ ) 4) √ t 5) f (t) = ⌊t⌋, the integer floor function
t t
sin t sin 5t
6) e−3t sin 5t sin 3t 7) e−t sin2 3t 8) e2t (3t5 − cos 4t) 9) sin t sin 2t sin 3t 10) t2 e−t cos t 11)
Z t Z t t t
1 − cos t −t e sin t
12) 2
13) e cos t dt 14) dt
t 0 0 t

Find the Laplace transform of the following functions:

2
√ 2 −t 3
(5t − 4 sinh 5t 2) (sin t − cos t) 3) sin( t 4) sinh 3t cos t 5) (1 + te )
1) 3 cosh
sin ωt, 0 < t < π/ω t, 0<t<π
6) f (t) = 7) f (t) = 8) t sin2 t 9) t2 cos at
0, π/ω < t < 2π/ω. π − t, π < t < 2π.
−t
sin t e sin t
10) 11)
t t
Find the inverse Laplace transform of the following functions:
3s + 7 1 − 7s 2s − 3 s (s + 2)2
1) 2) 3) 4) 5)
s2 − 2s − 3 (s − 3)(s − 1)(s + 2) s2 + 4s + 13 (s + 1)2 (s2 + 1) (s2+ 4s + 8)2
2 2

1 s +1  s  s + 2s + 3 1 1
6) 3
7) log 8) cot−1 9) 2 2
10) sin
s(s + a) s(s + 1) 2 (s + 2s + 2)(s + 2s + 5) s s

Using convolution theorem, find the inverse Laplace transform of the following functions:
1 s s 1 s
1) 2) 3) 4) 5)
s2 (s + 1)2 (s − 1)(s2 + 1) (s2 + a2 )(s2 + b2 ) (s2 + 4s + 13)2 (s + 2)(s2 + 9)

Solve the following differential equations by using Laplace transform method:

1. y ′′ + 4y ′ + 3y = e−t , y(0) = y ′ (0) = 1.

2. y ′′ − 3y ′ + 2y = e3t , when y(0) = 1 and y ′ (0) = 0.

3. (D2 − 3D + 2)y = 4e2t with y(0) = −3 and y ′ (0) = 5.


d2 y dy dy
4. 2
+ 2 − 3y = sin t, y = = 0 when t = 0.
dt dt dt
d3 y d2 y dy 2 2t dy d2 y
5. − 3 + 3 − y = t e , where y = 1, = 0, = −2 at t = 0.
dt3 dt2 dt dt dt2
6. y ′′ + y = 0, y(0) = 2, y( π2 ) = 1.
d4 y
7. 4
− k 4 y = 0, where y(0) = 1, y ′ (0) = y ′′ (0) = y ′′′ (0) = 0.
dt
8. ty ′′ + y ′ + 4ty = 0, where y(0) = 3, y ′ (0) = 0.

9. y ′′′′ (t) + 2y ′′ (t) + y(t) = sin t, when y(0) = y ′ (0) = y ′′ (0) = y ′′′ (0) = 0.

10. (D2 + 1)x = t cos 2t, x = Dx = 0 at t = 0.


d2 y dy
11. 2
+ 5 + 5y = e−t sin t, where y(0) = 0 and y ′ (0) = 1.
dt dt
12. (D2 + ω 2 )y = cos ωt, t > 0, given that y = 0 and Dy = 0 at t = 0.

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