Prepared By: Dr.S.

Anbarasu,MIT, Anna University

Unit -1 Ordinary Differential Equations

𝑑
1. Solve: (𝐷 3 − 2𝐷 2 − 𝐷 + 2)𝑦 = 0, 𝐷 ≡ 𝑑𝑡.
𝑑3 𝑥 𝑑2𝑥 𝑑𝑥
2. Solve: 𝑑𝑡 3 −3 𝑑𝑡 2 +3 𝑑𝑡 − 𝑥 = 0.
3. Find the particular integral of (𝐷 2 + 3𝐷 + 2)𝑦 = 5.
4. Find the particular integral of (𝐷 2 + 2𝐷 + 1)𝑦 = 𝑒 −𝑥 sin 𝑥.
5. Find the particular integral of (𝐷 2 + 6𝐷 + 9)𝑦 = 𝑒 −2𝑥 𝑥 3.
6. Find the particular integral of (𝐷 2 − 2𝐷)𝑦 = 𝑒 𝑥 cos 𝑥.
7. Find the particular integral of (𝐷 2 − 2𝐷 + 1)𝑦 = 𝑥 2.
8. Guess the trial solution of the particular integral for the differential equation
𝑦 ′′ − 𝑦 = 𝑒 𝑥 sin 2𝑥 using the method of undetermined coefficients.
9. By the method of undetermined coefficients, solve the differential equation
(𝐷 2 − 2𝐷)𝑦 = 5𝑒 𝑡 cos 𝑡.
10. Solve 𝑦 ′′ + 𝑦 = cos 𝑥 by method of undetermined coefficients.
11. Using method of undetermined coefficients solve 𝑦 ′′ + 𝑦 = 10 sin 𝑥.
12. Using method of variation of parameters solve the following differential
equation 𝑦 ′′ − 2𝑦 ′ + 𝑦 = 𝑥𝑒 𝑥 .
13. Using method of variation of parameters solve the following differential
equation 𝑦 ′′ + 4𝑦 = 4 tan 2𝑥.
𝑑2 𝑦
14. Solve 𝑑𝑥 2 + 𝑦 = cosec 𝑥 by using method of variation of parameter.
15. Find the Wronskian of the functions sin 2𝑥 & cos 2𝑥 and hence state
whether they are linearly independent or not.
16. By the method of variation of parameters, solve the differential equation
𝑒 3𝑡
(𝐷 2 − 6𝐷 + 9)𝑦 = .
𝑡2
𝑑2𝑦 𝑑𝑦
17. Solve: 𝑥 2 −𝑥 + 𝑦 = 0.
𝑑𝑥 2 𝑑𝑥
𝑑2𝑦 1 𝑑𝑦 12 log 𝑥
18. Solve: 𝑑𝑥 2 + 𝑥 𝑑𝑥 = .
𝑥2
19. Solve: (𝑥 2 𝐷 2 − 3𝑥𝐷 + 5)𝑦 = 𝑥 2 sin (log 𝑥).
20. Solve: (𝑥 2 𝐷 2 − 3𝑥𝐷 + 4)𝑦 = 𝑥 log 𝑥.
21. Reduce (2𝑥 + 3)2 𝑦 ′′ − 2(2𝑥 + 3)𝑦 ′ − 12𝑦 = 0 into a differential equation
with constant coefficients.(Do not solve it).
22. Solve: (𝑥 + 1)2 𝑦 ′′ + (𝑥 + 1)𝑦 ′ + 𝑦 = 2 sin [log(1 + 𝑥)].
23. Solve: (𝐷 − 7)𝑥 + 𝑦 = 0; −2𝑥 + (𝐷 − 5)𝑦 = 0.
24. Solve : 𝐷𝑥 − (𝐷 − 2)𝑦 = cos 2𝑡; (𝐷 − 2)𝑥 + 𝐷𝑦 = sin 2𝑡.
25. Solve: 𝐷𝑥 + 𝑦 = sin 𝑡 ; 𝑥 + 𝐷𝑦 = cos 𝑡, given that 𝑥 = 2 and 𝑦 = 0 at 𝑡 = 0.
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

Unit-2 – Laplace Transforms

1. Find the Laplace transform of 𝑓(𝑡) = 𝑡 cosh 𝑡.

2. State the sufficient conditions for the existence of Laplace transform.
𝑒 −𝑡 −𝑒 2𝑡
3. Find the Laplace transform of .
𝑡
4. Find the Laplace transform of 𝑓(𝑡) = 𝑡𝑒 −𝑡 sin 𝑡.
5. If 𝑓(𝑡) = 𝑒 −2𝑡 sin 2𝑡, find L{𝑓′(𝑡)}.
𝑡
6. Evaluate ∫0 𝑒 −𝑡 sin 𝑡 𝑑𝑡 .
∞ sin2 𝑡
7. Using Laplace transform, evaluate ∫0 𝑑𝑡 .
𝑡𝑒 𝑡
∞
8. Evaluate ∫0 𝑡𝑒 −3𝑡 sin 𝑡 𝑑𝑡
9. Find the Laplace transform of the periodic function defined on the
𝜋
2𝜋
sin 𝜔𝑡 , 0<𝑡<𝜔
interval 0 < 𝑡 < by 𝑓(𝑡) = { 𝜋 2𝜋 and
𝜔 0, <𝑡<
𝜔 𝜔
2𝜋
𝑓 (𝑡 + ) = 𝑓(𝑡).
𝜔
10. Find the Laplace transform of the square wave function
𝑎
1, 0<𝑡<2
𝑓(𝑡) = { 𝑎 , where 𝑓(𝑡 + 𝑎) = 𝑓(𝑡).
−1, <𝑡<𝑎
2
1
11. If 𝐹(𝑠) = 𝑠(𝑠+𝑎) is the Laplace transform of a function 𝑓(𝑡), then find the
value of lim 𝑓(𝑡) without finding the function 𝑓(𝑡).
𝑡→∞
12. Verify initial and final value theorems for the function
𝑓(𝑡) = 1 + 𝑒 −𝑡 (sin 𝑡 + cos 𝑡).
13. Using initial value theorem, find lim 𝑓(𝑡), given that
𝑡→0
1
𝐿{𝑓(𝑡)} = 𝐹(𝑠) = .
(𝑠+1)2 +1

14. Use the first shifting property to find the inverse Laplace transform of
1
.
(𝑠+3)4
1
15. Find the inverse Laplace transform of (𝑠2 +2𝑠+2).
𝑠+1
16. Find the inverse Laplace transform of log ( ).
𝑠
5𝑠+3
17. Find the inverse Laplace transform of (𝑠−1)(𝑠2 .
+2𝑠+5)
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

18. Use convolution theorem, to find the inverse Laplace transform of

1
(𝑠+1)(𝑠 2 +1)
.
1
19. Using convolution theorem, find the inverse Laplace transform of 𝑠(𝑠+𝑎)3.
1
20. Using convolution theorem, find the inverse Laplace transform of (𝑠2 +9)2.
𝑠
21. Using convolution theorem, find the inverse Laplace transform of (𝑠2 +4)2
22. Use Laplace transform method to solve the differential equation 𝑦 ′′ +
𝑦 = 2 given that 𝑦(0) = 0, 𝑦 ′ (0) = 1.
23. Solve 𝑦 ′′ − 4𝑦 ′ + 3𝑦 = 𝑒 −𝑡 ; 𝑦(0) = 1, 𝑦 ′ (0) = 0 by Laplace transform
method.
24. Solve 𝑦 ′′ − 2𝑦 ′ + 𝑦 = 𝑡𝑒 −𝑡 ; 𝑦(0) = 1, 𝑦 ′ (0) = 1 by Laplace transform
method.
25. Solve using Laplace transform
𝑑2 𝑦 𝑑𝑦
+ 4 𝑑𝑡 + 4𝑦 = sin 𝑡 ; 𝑦(0) = 2, 𝑦 ′ (0) = 0.
𝑑𝑡 2
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

Unit-3 – FOURIER SERIES

1. State Dirichlet’s conditions for existence of Fourier series.

𝜋2 cos 𝑛𝑥
2. If (𝜋 − 𝑥)2 = + 4 ∑∞
𝑛=1 in 0 < 𝑥 < 2𝜋, then deduce the value
3 𝑛2
(−1)𝑛+1
of ∑∞
𝑛=1 .
𝑛2
𝑙 − 𝑥, −𝜋 < 𝑥 < 0
3. If 𝑓(𝑥) = { , then find the constant term of Fourier
𝑙 + 𝑥, 0 < 𝑥 < 𝜋
series expansion of 𝑓(𝑥).
4. Find the Fourier series expansion of 𝑓(𝑥) = 𝑥 2 in, (−𝜋, 𝜋) of periodicity
2𝜋.
5. Define the root mean square values of the function 𝑓(𝑥) in, (0,2𝜋).
6. State Parseval’s theorem.
7. If the half-range sine series of 𝑥(𝜋 − 𝑥) in 0 ≤ 𝑥 ≤ 𝜋 is ∑ 𝑏𝑛 sin 𝑛𝑥, find
the value of ∑ 𝑏𝑛2 .
8. Find the Fourier series expansion of 𝑓(𝑥) = 𝑥 2 − 𝑥 in, (−𝜋, 𝜋) of
1
periodicity 2𝜋. Hence deduce the sum of the series ∑∞
𝑛=1 𝑛4, assuming
1 𝜋2
that ∑∞
𝑛=1 𝑛2 = .
6
9. Find the Fourier series expansion of 𝑓(𝑥) = |cos 𝑥| in, (−𝜋, 𝜋) of
periodicity 2𝜋.
10. Find the Fourier series expansion of 𝑓(𝑥) = sinh 𝑎𝑥 in, (−𝜋, 𝜋) of
periodicity 2𝜋.
11. Find the Fourier series expansion of 𝑓(𝑥) = 𝑥 2 + 𝑥 in, (−2,2) . Hence
1
find the sum of the series ∑∞
𝑛=1 2. 𝑛
−𝜋, −𝜋 < 𝑥 < 0
12. Obtain the Fourier series expansion of 𝑓(𝑥) = { .
𝑥, 0<𝑥<𝜋
1, 0 < 𝑥 < 𝜋
13. Obtain the Fourier series expansion of 𝑓(𝑥) = { . Hence
2, 𝜋 < 𝑥 < 2𝜋
1
find the sum of the series ∑∞
𝑛=0 (2𝑛+1)2.

14. Obtain the half range Fourier sine series of 𝑓(𝑥) = 𝑥 in 0 < 𝑥 < 𝑙.
1 𝜋2
Hence deduce that ∑∞
𝑛=1 𝑛2 = .
6
15. Obtain the half range Fourier cosine series of 𝑓 (𝑥 ) = 𝑥 sin 𝑥 in 0 < 𝑥 <
1 1 1
𝜋. Hence deduce the value of 1 . 3 − + − ⋯ to ∞.
3. 5 5. 7
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

16. Obtain the half range Fourier cosine series of 𝑓(𝑥) = (𝑥 − 1)2 in 0 <
𝑥 < 𝑙.
17. Find the half range Fourier cosine series of 𝑓(𝑥 ) = (𝜋 − 𝑥)2 in 0 < 𝑥 <
1
𝜋. Hence find the sum of the series ∑∞
𝑛=1 𝑛4.

18. Find the Fourier series of period 2 for the function 𝑓(𝑥) =
𝑘, −1 < 𝑥 < 0 1 1 1
{ . Hence find the sum of 1- 3 + 5 - 7 +.. ∞ and
𝑥, 0 < 𝑥 < 1
1
∑∞ 𝑛=0 (2𝑛+1)2.

19. Compute the first two harmonics of Fourier series expansion of 𝑓(𝑥)
from the table give below:
𝑥 0 𝜋 2𝜋 𝜋 4𝜋 5𝜋 𝜋
3 3 3 3
𝑓(𝑥) 1.0 1.4 1.9 1.7 1.5 1.2 1.0
20. The following table gives the variations of periodic current over a period
𝑡(secs) 0 𝑇 𝑇 𝑇 2𝑇 5𝑇 𝑇
6 3 2 3 6
𝐴 (amp) 1.98 1.30 1.05 1.30 - 0.88 -0,25 1.98
Express 𝐴 in a Fourier series as far as the second harmonic using the
above data and also obtain the amplitude of the first harmonic.
21. Obtain the constant term and the first three harmonics in the Fourier
cosine series of 𝑦 = 𝑓(𝑥) in (0,6) using the following table:
𝑥: 0 1 2 3 4 5
𝑦: 4 8 15 7 6 2
22. State True or False: Fourier series of the period 2𝜋 for the function
𝑓(𝑥) = 𝑥 cos 𝑥 in the interval (−𝜋, 𝜋) contains only sine terms. Justify
your answer.
23. Define the complex form of 𝑓(𝑥) in, (𝑐, 𝑐 + 2𝑙).
24. Find the complex form of the Fourier series of the periodic function
(𝑥 ) = cos 𝑎𝑥 , −𝜋 < 𝑥 < 𝜋 .
25. Find the complex form of 𝑓(𝑥) = 𝑒 −𝑥 ,when 0 < 𝑥 < 𝑙
and 𝑓(𝑥 + 2) = 𝑓(𝑥).
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

Unit-4 – FOURIER TRANSFORMS

1. If the Fourier transform of 𝑓(𝑥) is 𝐹(𝑠), then find the Fourier
transform of 𝑒 𝑖𝑎𝑥 𝑓(𝑥).
−𝑑
2. Show that 𝐹𝑠 {𝑥𝑓(𝑥)} = [𝐹𝑐 (𝑠)], where 𝐹𝑠 and 𝐹𝑐 are Fourier sine
𝑑𝑠
and cosine transforms respectively.
1 − 𝑥 2 , 𝑖𝑓|𝑥| < 1
3. Find the Fourier transform of 𝑓(𝑥) = { .
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
4. Find the Fourier cosine transform of 𝑒 𝑎𝑥 , 𝑎 > 0, hence show that
∞ cos 𝑠𝑥 𝜋
∫0 𝑑𝑥 = 𝑒 −𝑎𝑠 .
𝑥 2 +𝑎2 2𝑎
2𝑥2
5. Obtain the Fourier transform of 𝑓(𝑥) = 𝑒 −𝑎 , for 𝑎 > 0, hence find
𝑥2
the Fourier transform of 𝑒 − 2 .
∞ 𝑥 2 𝑑𝑥
6. By obtaining Fourier sine transform 𝑒 −𝑎𝑥 , 𝑎 > 0, evaluate ∫0 .
(𝑥 2 +16)2
7. State and prove modulation theorems on Fourier transforms.
∞
8. Solve for 𝑓(𝑥), if ∫0 𝑓(𝑥) cos 𝛼𝑥 𝑑𝑥 = 𝑒 −𝛼 , where 𝛼 > 0.
𝑎 − |𝑥|, 𝑓𝑜𝑟 |𝑥| < 𝑎
9. Find the Fourier transform of 𝑓(𝑥) = { .
0, 𝑓𝑜𝑟 |𝑥| > 𝑎 > 0
∞ sin2 𝑡
Hence deduce the value of ∫0 𝑑𝑡 .
𝑡2
∞ 𝑠2
10. Evaluate ∫0 𝑑𝑠 using suitable Fourier transform
(𝑠 2 +𝑎2)(𝑠2 +𝑏2 )
technique.
11. Verify convolution theorem for Fourier transform if
2
𝑓(𝑥) = 𝑔(𝑥) = 𝑒 −𝑥 .
12. State Fourier integral theorem.
1, 𝑓𝑜𝑟 |𝑥 | ≤ 1
13. Find the Fourier transform of 𝑓(𝑥 ) = { .
0, 𝑓𝑜𝑟 |𝑥| > 1
1
14. Find the cosine Fourier transform of 𝑓(𝑥) = .
1+𝑥 2
15. Using Fourier integral representation show that
∞ 𝜔 sin 𝑥𝜔 𝜋
∫0 𝑑𝜔 = 2 𝑒 −𝑥 for 𝑥 > 0.
1+𝜔2
16. If the Fourier transform of 𝑓(𝑥) is 𝐹(𝑠), then prove that
1
𝐹 {𝑓(𝑥) cos 𝑎𝑥} = {𝐹 (𝑆 + 𝑎) + 𝐹(𝑆 − 𝑎)}.
2
17. Find the Fourier cosine transform of 𝑓(𝑥) = 5𝑒 −2𝑥 .
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

18. Find the Fourier sine and cosine transform of 𝑒 −𝑥 and hence find the
𝑥 1
Fourier sine transform of 1+𝑥 2 and Fourier cosine transform of 1+𝑥 2.
𝑠
19. Given that 𝐹𝑠 {𝑓(𝑥)} = 𝑠 2 +𝑎2 for 𝑎 > 0, hence find 𝐹𝐶 {𝑥𝑓(𝑥)}.
1
20. Find the Fourier cosine transform of 𝑓(𝑥) = .
√𝑥
1
21. Find the Fourier sine transform of 𝑓(𝑥) = 𝑥.
∞ 𝑥 2 𝑑𝑥
22. Using Parseval’s identity evaluate ∫0 (𝑥 2 +𝑎2)2
.
23. Find the Fourier sine transform of
sin 𝑥, 0 ≤ 𝑥 ≤ 𝑎
𝑓(𝑥) = { .
0, 𝑥>𝑎
24. Find the Fourier sine and cosine transforms of 𝑥 𝑛−1 .
sin 𝑎𝜆
25. Find the function 𝑓(𝑥) if its cosine transform is given by 𝜆
.
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

Unit -5 – Z-Transforms

1. Find the Z-transform of 𝑛2𝑛 .

1
2. Find the Z-transform of 𝑛!.
3. State initial and final value theorems for Z-transform.
4. If 𝑍{𝑓(𝑛)} = 𝐹(𝑧), then prove that 𝑍{𝑓(𝑛 − 𝑘)} = 𝑧 −𝑘 𝐹(𝑧), where 𝑘 is
a positive integer.
1
5. Find the Z-transform of 𝑛(𝑛+1), for 𝑛 ≥ 1.
𝑧3
6. Using method of partial fraction find 𝑍 −1 [(𝑧−1)2 (𝑧−2)].
𝑧3
7. Find the inverse Z-transform of (𝑧−𝑎)3 by using convolution theorem.
𝑧2
8. Using convolution theorem, find 𝑍 −1 [(𝑧−1)(𝑧−3)].
𝑧2
9. Using convolution theorem, find 𝑍 −1 [ 1 1 ].
(𝑧− 2)(𝑧− 4)
𝑧2
10. Using convolution theorem, evaluate 𝑍 −1 [(𝑧 2 −5𝑧+6)].
𝑧(𝑧 2 −𝑧+2)
11. Find the inverse Z-transform of (𝑧+1)(𝑧−2)2 , by using residue
theorem.
2𝑧 2 +3𝑧+12
12. If 𝑍{𝑓(𝑛)} = (𝑧−1)4
, then find 𝑓(2) and 𝑓(3), by using initial
value theorem.
13. Using Z-transform, solve 𝑢𝑛+2 + 3𝑢𝑛+1 + 2𝑢𝑛 = 0,
given 𝑢0 = 1, 𝑢1 = 2.
14. Solve the difference equation using Z-transform technique
𝑦𝑛+2 + 4𝑦𝑛+1 + 3𝑦𝑛 = 2𝑛 , with 𝑦0 = 0, 𝑦1 = 1.
𝑧 2 +𝑧
15. Using inversion integral, find the inverse Z-transform of .
(𝑧−1)(𝑧 2 +1)
2𝑧 2 +5𝑧+14
16. If 𝑈(𝑧) = (𝑧−1)4
is the Z-transform of the sequence 𝑢𝑛 , find 𝑢2 and
𝑢3 .
17. Find the Z-transform of cos(𝑛𝜃) and hence show that
𝑧(𝑧−𝑎 cos 𝜃)
𝑍{𝑎𝑛 cos(𝑛𝜃)} = 𝑧 2 −2𝑎𝑧 cos 𝜃+𝑎2 .
18. Solve, by using Z-transform 𝑦𝑛+2 + 𝑦𝑛 = 2, given that 𝑦0 = 𝑦1 = 0.
19. From the relation 𝑦𝑛 = 𝑎2𝑛 + 𝑏(−2)𝑛 , derive a difference equation by
eliminating the constants 𝑎 and 𝑏.
 Prepared By: Dr.S. Anbarasu,MIT, Anna University

20. Solve the difference equation 𝑦(𝑛 + 3) − 3𝑦(𝑛 + 1) + 2𝑦(𝑛) = 0

given that 𝑦(0) = 4, 𝑦(1) = 0 and 𝑦(2) = 8.
21. Find the Z-transform of the sequence {4,8,16,32,….}.
𝑧
22. Find the inverse Z-transform of (𝑧 2 −3𝑧+2) by long division method.
𝑧
23. Find 𝑢𝑛 if 𝑈(𝑧) = log 𝑧+1.
24. Evaluate ∑𝑛𝑘=1 𝑘 2 using Z-transform.
2𝑛 , 𝑛 < 0
25. Find the Z-transform of 𝑓(𝑛) = { 𝑛 .
4 , 𝑛≥0