2022-2023
Sessional Test I– April, 2023
Semester II
ID No: …………………………
[Total No. of Pages: 01]
Time: 90 minutes
Max. Marks: 40
Title of the Course: Differential Equations and Transformations
Course Code: 22AS002
Instructions:
For Section A
• There are 5 questions of 1 marks each. Each question is having four distinct options out of which only one choice
will be correct. There is no negative marking for incorrect answers.
For Section B
 There are 5 Questions of 3 marks each. All are compulsory.
For Section C
 There are 2 Questions of 5 marks each. All are compulsory.
For Section D
 There is 1 Question of 10 marks and compulsory.
__________________________________________________________________________________
Section-A
(All Questions are Compulsory, Each question carries 01 mark)
1. Which of the following functions is an even function in the interval [−𝜋, 𝜋]?
(a) x
(b) x 3 +sin 𝑥
3
(c) x + cos 𝑥
(d) x 3 sin 𝑥
2. In the Fourier cosine series expansion of a periodic function 𝑓(𝑥), 0 < 𝑥 < 𝜋 , the coefficient 𝑎𝑛 is given by
0
(a)
(c)
1 𝜋
∫ 𝑓(𝑥) sin 𝑛𝑥 𝑑𝑥
𝜋 0
(b)
2 𝜋
∫ 𝑓(𝑥) cos 𝑛𝑥 𝑑𝑥
𝜋 0
(d)
1 𝜋
∫ 𝑓(𝑥) cos 𝑛𝑥 𝑑𝑥
𝜋 0
3. Which of the following is not a solution to the differential equation
(a) sin 𝑡
(b)
(c) sin 𝑡 + cos 𝑡
4. An integrating factor for the differential equation
(a)
(c)
𝑑2 𝑥
𝑑𝑡 2
+𝑥 =0
cos 𝑡
(d) tan 𝑡
(y 4
+ 2𝑦)𝑑𝑥 + (𝑥𝑦 3 + 2𝑦 4 − 4𝑥)𝑑𝑦 = 0 is
1
𝑦3
(b) 𝑦
1
1
𝑥2
(d) x 2
5. The complementary function for the differential equation 𝑦 ′′ − 2𝑦 ′ + 𝑦 = 0 is
(a) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥
(b) 𝑐1 𝑒 𝑥 + 𝑐2 𝑥𝑒 𝑥
(c) 𝑐1 𝑒 𝑥 + 𝑐2 𝑥𝑒 −𝑥
(d) 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥
1
Section-B
(Attempt all questions, each question carries 03 marks)
6. . In the Fourier series expansion of the periodic function 𝑓(𝑥) = 𝑥 , 0 < x < 2π the coefficient of sin 2𝑥 is?
7. Solve : ( 𝑥 𝑦 3 + 𝑦 )𝑑𝑥 + 2(𝑥 2 𝑦 2 + 𝑥 + 𝑦 4 )𝑑𝑦 = 0.
8. Solve : 𝑦 + 𝑝𝑥 = 𝑥 4 𝑝2.
𝑑2 𝑦
9. Solve by method of variation of parameters the differential equation 𝑑𝑥 2 + 𝑦 = 𝑠𝑒𝑐𝑥
10..If 𝑓(𝑥) = {
𝑥,
0<𝑥<
𝜋
2
𝜋 − 𝑥,
𝜋
2
<𝑥< 𝜋
. Show that 𝑓(𝑥) =
4
𝜋
[ 𝑠𝑖𝑛𝑥 −
𝑠𝑖𝑛3𝑥
32
+
𝑠𝑖𝑛5𝑥
52
− ⋯ … … ].
Section-C
(Attempt all questions, each question carries 5 marks)
11. Represent the following function by a Fourier series:
12. The initial value problem governing the current ‘i’ flowing in series R. L. circuit when a voltage v(t) = t is applied
𝑑𝑖
is given by 𝑖 𝑅 + 𝐿 𝑑𝑡 = 𝑡; 𝑡 ≥ 0; 𝑖(0) = 0. Where R, L are constants. Find the current i(t) at time t.
Section-D
(Attempt all questions, each question carries 10 marks)
13. (a) Find the fourier series for the function f(x) is given by
𝑓(𝑥) = {
1+
1−
2𝑥
𝜋
2𝑥
𝜋
−𝜋 ≤𝑥 ≤0
0 ≤𝑥 ≤ 𝜋
1
Hence deduce that 12 +
(b) Solve 𝑥 3
𝑑3 𝑦
𝑑𝑥 3
+ 2 𝑥2
1
32
+
𝑑2 𝑦
𝑑𝑥 2
1
52
+ ⋯…… =
𝜋2
8
+ 2𝑦 = 10 ( 𝑥 +
1
𝑥
)
2