4500ICBTEG: Engineering Mathematics
Page 1 of 2
Question No. 1 [30 Marks]
1.1 Find the particular solution of the following differential equation
𝑑𝑦 𝑑𝑥
𝑥
+ 2𝑦 + 3𝑥 = 0
Hence find the value of y when x= 2+ 𝛽
𝑦(−1 + 𝛽) = 3 + 𝛼
[13 marks]
1.2
a) Show that the following differential equation is homogenous
2+
(𝑥𝑦 + 𝑥2) 𝑑𝑦 = 0
𝑦
𝑑𝑥
b) By using a suitable substitution find the general solution of the above equation
[17 marks]
Question No. 2 [25 Marks]
2.1 Find the general solution of the following second order linear differential equation
𝑑2𝑦2
𝑑𝑦𝑑𝑥
2𝑦
=0
− 2𝑚+ 𝑚
𝑚∈𝑍
𝑑𝑥
[5 marks]
2.2 Let 𝑢(𝑥, 𝑦) = (𝛽 + 2)𝑥𝑒8(𝑥−𝑦). Find the value of (3 + 𝛼) 𝜕𝑢
− 2(𝛽 + 2) 𝜕𝑢
𝜕𝑥
+ 𝑢 when x=1+ 𝛼 and
𝜕𝑦
y=1+ 𝛽
[ 8 marks]
2.3 Find the rate of change of 𝐻(𝑥, 𝑦) correct to 4 significant figures when x is 3+ 𝛼 units and y is 2+ 𝛽 units.
x is decreasing at 6+ 𝛼 units/s and y is increasing at 3.5+ 𝛽 units/s.
𝐻(𝑥, 𝑦) = (5 + 𝛼)𝑒2𝑦𝑥3
Nov 2021 – March 2022
4500ICBTEG: Engineering Mathematics
Page 2 of 2
[12 marks]
Question No. 3 [25 Marks]
3.1. Motion of a particle at time t is given by
𝑥 = (2 + 𝛼)𝑡3 + 1 𝑦 = −𝑡3 + (2 + 𝛽)𝑡 and 𝑧 = 2𝑡2 −
(3 + 𝛼)𝑡
Find the components of acceleration in the direction of the vector 2𝑖 + (3 + 𝛼)𝑗 − 2(𝛽 + 1)𝑘 when t=2+𝛼
[11 marks]
3.2. A vector field ⃗𝑨⃗⃗ (𝑥, 𝑦, 𝑧) is given by ⃗𝑨⃗⃗ = 𝑥3𝑦2 𝒊 + (2 + 𝛼)𝑦𝑧2𝒋 + 𝑥2𝑧𝒌
a) Find div A at the point (2+ 𝛽,-1, 1+ 𝛼)
[5 marks]
b) Curl A at the point (1+ 𝛽,-2,1+ 𝛼)
[9 marks]
Question No. 4 [20 Marks]
4.1 Using the standard list of the Laplace transforms and properties, Find the Laplace transform of the
following functions
a) 𝑓(𝑡) = 𝐶𝑜𝑠2(3 + 𝛼)𝑡 b) 𝑓(𝑡) = 𝑡𝑒3𝑡𝐶𝑜𝑠(2 + 𝛽)𝑡
[12 marks]
4.2. Determine 𝐿−1 [
𝛼]
𝑠
[8 marks]
*** End of the Paper ***
Nov 2021 – March 2022