Practice Problems for All of the Topics of Engineering Mathematics

Department of Mathematics
15B11MA211 Mathematics-2
B.Tech. Core

Tutorial Sheet 1 (Differential Equations with Constant Coefficients)

1. Find the complementary function of the following equations:
i. (𝐷2 − 2𝐷 + 2)𝑦 = 0
ii. (𝐷4 − 81)𝑦 = 0
iii. (𝐷3 − 1)2 𝑦 = 0

2. Solve the following differential equations.
i. (𝐷2 − 4𝐷 + 4)𝑦 = 𝑒 2𝑥 + 𝑥 3 + 𝑐𝑜𝑠 2 𝑥
ii. (𝐷2 − 6𝐷 + 13)𝑦 = 16 𝑒 3𝑥 𝑠𝑖𝑛 4 𝑥 + 3𝑥
iii. (𝐷2 + 1)𝑦 = 𝑐𝑜𝑠𝑒𝑐 𝑥

3. Find the solution of the following differential equations:
𝑑2𝑦 𝑑𝑦
i. 𝑥2 2 + 𝑥 + 𝑦 = 0
𝑑𝑥 𝑑𝑥
ii. 𝑥 𝑦 + 4𝑥𝑦 ′ + 2𝑦 = 0
2 ′′

iii. 𝑥 2 𝑦 ′′ − 5𝑥𝑦 ′ + 9𝑦 = 0
iv. 𝑥 3 𝑦 ′′′ + 3𝑥 2 𝑦 ′′ + 𝑥𝑦 ′ + 𝑦 = sin(𝑙𝑜𝑔𝑥) + 𝑥


Ans:
1.
i. 𝑦(𝑥) = 𝑒 𝑥 (𝐴cos 𝑥 + B 𝑠𝑖𝑛 𝑥)
ii. 𝑦(𝑥) = 𝐴𝑒 3𝑥 + 𝐵𝑒 −3𝑥 + C cos 3𝑥 + D 𝑠𝑖𝑛 3𝑥
√3 √3
iii. 𝑦(𝑥) = (𝐴 + 𝐵𝑥) 𝑒 𝑥 + 𝑒 −𝑥/2 ( (C + Dx) cos ( 2 𝑥) + (E + Fx) sin ( 2 𝑥))
2.
𝑥2 𝑥3 3𝑥 2 9𝑥 3 sin 2𝑥
i. 𝑦(𝑥) = (𝐴 + 𝐵𝑥)𝑒 2𝑥 + 𝑒𝑥+ + + + -
2 4 4 8 4 8
4 3𝑥 3𝑥
ii. 𝑦(𝑥) = 𝑒 3𝑥 (𝐴cos 2𝑥 + B 𝑠𝑖𝑛 2𝑥) − 3 𝑒 sin 4𝑥 + (log 3)2 −6 log 3+13
iii. y(x)=(𝐴 − 𝑥) cos 𝑥 + (𝐵 + log | sin 𝑥|) sin 𝑥

3.
i. 𝑦(𝑥) = 𝐴𝑐𝑜𝑠(𝑙𝑜𝑔𝑥) + 𝐵𝑠𝑖𝑛(𝑙𝑜𝑔𝑥)
𝐴 𝐵
ii. 𝑦(𝑥) = 𝑥 + 𝑥 2
iii. 𝑦(𝑥) = (𝐴 + 𝐵𝑙𝑜𝑔𝑥)𝑥 3
𝐴 √3 √3 sin(𝑙𝑜𝑔𝑥)+cos(𝑙𝑜𝑔𝑥)+𝑥
iv. 𝑦(𝑥) = 𝑥 + 𝑥1/2 (𝐵cos ( 2 𝑙𝑜𝑔𝑥) + C 𝑠𝑖𝑛 ( 2 𝑙𝑜𝑔𝑥)) + 2

,