SIES Graduate School of Technology

Nerul, Navi Mumbai

First Year Engineering

Applied
Mathematics -II
Handbook
Semester-II
Academic year: 2024-25
[As per the University of Mumbai syllabus R2025]
 SIES Graduate School of Technology

Question Bank

Sr.
Problem Solution
No.
Module 01. Exact Differential Equation
2
1 Solve
2 2 𝑒 𝑥𝑦 + 𝑥 4 = 𝑦 3 = 𝑐
(𝑦 2 𝑒 𝑥𝑦 + 4𝑥 3 )𝑑𝑥 + (2𝑥𝑦𝑒 𝑥𝑦 − 3𝑦 2 )𝑑𝑦 = 0
1
2 Solve {𝑦 (1 + ) + 𝑐𝑜𝑠𝑦} 𝑑𝑥 + (𝑥 + 𝑙𝑜𝑔𝑥 − 𝑥𝑠𝑖𝑛𝑦)𝑑𝑦 = 0 𝑦(𝑥 + 𝑙𝑜𝑔𝑥) + 𝑥𝑐𝑜𝑠𝑦 = 𝑐
𝑥
3 Solve (1 + 2𝑥𝑦𝑐𝑜𝑠𝑥 − 2𝑥𝑦)𝑑𝑥 + 2 (𝑠𝑖𝑛𝑥 2 −𝑥 2 )𝑑𝑦
=0 𝑥 + 𝑦𝑠𝑖𝑛𝑥 2 − 𝑥 2 𝑦 = 𝑐
𝑑𝑦 𝑦𝑐𝑜𝑠𝑥+𝑠𝑖𝑛𝑦+𝑦
4 Solve
𝑑𝑥
+
𝑠𝑖𝑛𝑥+𝑥𝑐𝑜𝑠𝑦+𝑥
=0 𝑦𝑠𝑖𝑛𝑥 + 𝑥𝑠𝑖𝑛𝑦 + 𝑥𝑦 = 𝑐
5 Solve (𝑥 2 − 4𝑥𝑦 − 2𝑦 2 )𝑑𝑥
+ (𝑦 2 − 4𝑥𝑦 − 2𝑥 2 )𝑑𝑦 = 0 𝑥 3 + 𝑦 3 − 6𝑥 2 𝑦 − 6𝑦 2 𝑥 = 𝑐
6 Solve [𝑥√𝑥 2 + 𝑦 2 − 𝑦]𝑑𝑥 + [𝑦√𝑥 2 + 𝑦 2 − 𝑥]𝑑𝑦 = 0 (𝑥 2 + 𝑦 2 )3/2 − 3𝑥𝑦 = 𝑐
7 (𝑡𝑎𝑛𝑦 + 𝑥)𝑑𝑥 + (𝑥𝑠𝑒𝑐 2 𝑦 − 3𝑦)𝑑𝑦 = 0 2𝑥𝑡𝑎𝑛𝑦 − 3𝑦 2 + 𝑥 2 = 𝑐
8 (𝑥 − 2𝑒 𝑦 )𝑑𝑦 + (𝑦 + 𝑥𝑠𝑖𝑛𝑥)𝑑𝑥 = 0 𝑥𝑦 − 𝑥𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥 − 2𝑒 𝑦 = 𝑐

9 (4𝑥 + 3𝑦 − 4)𝑑𝑥 + (3𝑥 − 7𝑦 − 3)𝑑𝑦 = 0 4𝑥 2 + 6𝑥𝑦 − 8𝑥 − 7𝑦 2 − 6𝑦 = 𝑐

10 (𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 + 𝑒 2𝑥 )𝑑𝑥 + (𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦 + 𝑡𝑎𝑛𝑦)𝑑𝑦 = 0 1

−𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 + 𝑒 2𝑥 + 𝑙𝑜𝑔𝑠𝑒𝑐𝑦 = 𝑐
2
𝝏𝑴 𝝏𝑵
Type I: If ( 𝝏𝒚 − 𝝏𝒙 ) /𝑵 is a function of x, 𝒇(𝒙)𝒔𝒂𝒚 𝒕𝒉𝒆𝒏 𝑰. 𝑭 𝒊𝒔 𝒆∫ 𝒇(𝒙)𝒅𝒙
1 1 1/𝑥 3 𝑦2
11 Solve (𝑥𝑦 2 − 𝑒 𝑥3 ) 𝑑𝑥 − 𝑥 2 𝑦𝑑𝑦 = 0 3
𝑒 − 2 𝑥2 = 𝑐
12 Solve (𝑥 2 + 𝑦 2 + 1)𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 𝑥−
𝑦2 1
−𝑥 =𝑐
𝑥
13 Solve (2𝑥𝑙𝑜𝑔𝑥 − 𝑥𝑦)𝑑𝑦 + 2𝑦𝑑𝑥 = 0 𝑦2
2𝑦𝑙𝑜𝑔𝑥 − =𝑐
2
14 Solve (4𝑥𝑦 + 3𝑦 2 − 𝑥)𝑑𝑥 + 𝑥(𝑥 + 2𝑦)𝑑𝑦 = 0 4𝑥 2 𝑦 + 4𝑥 3 𝑦 2 − 𝑥 4 = 𝑐
15 Solve (𝑥 4 + 𝑦 4 ) − 𝑥𝑦 3 𝑑𝑦 = 0 4𝑥 4 𝑙𝑜𝑔𝑥 − 𝑦 4 = 𝑐𝑥 4
𝝏𝑵 𝝏𝑴
Type II: If ( 𝝏𝒙 − ) /𝑴 is a function of 𝒚 𝒇(𝒚)𝒔𝒂𝒚 𝒕𝒉𝒆 𝑰. 𝑭 𝒊𝒔 𝒆∫ 𝒇(𝒚)𝒅𝒚
𝝏𝒚
16 Solve
(𝑥+2𝑦 3 )𝑑𝑦
=𝑦 𝑥 − 𝑦 3 = 𝑐𝑦
𝑑𝑥
𝑥 )𝑑𝑥
17 Solve 𝑦(𝑥𝑦 + 𝑒 − 𝑒 𝑥 𝑑𝑦 = 0 𝑥2 𝑒 𝑥
+ =𝑐
2 𝑦
18 𝑦 𝑦𝑙𝑜𝑔𝑥 − 𝑥𝑠𝑖𝑛𝑦 = 𝑐
Solve (𝑥 𝑠𝑒𝑐𝑦 − 𝑡𝑎𝑛𝑦) 𝑑𝑥 + (𝑠𝑒𝑐𝑦𝑙𝑜𝑔𝑥 − 𝑥)𝑑𝑦 = 0
19 Solve (𝑥𝑦 3 + 𝑦)𝑑𝑥 + 2(𝑥 2 𝑦 2 + 𝑥 + 𝑦 4 )𝑑𝑦 = 0 3𝑥 2 𝑦 4 + 6𝑥𝑦 2 + 𝑦 6 = 𝑐
Type III: I.F. for an equation of the type 𝒇𝟏 (𝒙𝒚)𝒚𝒅𝒙 + 𝒇𝟐 (𝒙𝒚)𝒙𝒅𝒚 = 𝟎:
𝟏
If 𝑴𝒅𝒙 + 𝑵𝒅𝒚 = 𝟎 is of the above form then is an IF provided that 𝑴𝒙 − 𝑵𝒚 ≠ 𝟎.
𝑴𝒙−𝑵𝒚
17 Solve 𝑦(𝑥𝑦 + 2𝑥 2 𝑦 2 )𝑑𝑥 + 𝑥(𝑥𝑦 − 𝑥 2 𝑦 2 )𝑑𝑦 = 0 𝑥2 1
log ( )= +𝑐
𝑦 𝑥𝑦
18 Solve (3𝑥 2 𝑦 4 + 2𝑥𝑦)𝑑𝑥 + (2𝑥 3 𝑦 3 − 𝑥 2 )𝑑𝑦 = 0 𝑥 3 𝑦 3 + 𝑥 2 = 𝑐𝑦
19 𝑑𝑦 𝑥 2 𝑦 3 +2𝑦 1 𝑥 1
Solve 𝑑𝑥 = − 2𝑥−2𝑥 3𝑦 2 log ( 2 ) − 2 2 = 𝑐
3 𝑦 3𝑥 𝑦
20 Solve (𝑦 − 𝑥𝑦 2 )𝑑𝑥 − (𝑥 + 𝑥𝑦 2 )𝑑𝑦 = 0 𝑥
log = 𝑐 + 𝑥𝑦
𝑦

1|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

21 Solve 𝑦(𝑥 + 𝑦)𝑑𝑥 − (𝑦 − 𝑥)𝑥𝑑𝑦 = 0 𝑥

log = 𝑐 + 𝑥𝑦
𝑦
Type IV: I.F. of an homogeneous: If 𝑴𝒅𝒙 + 𝑵𝒅𝒚 = 𝟎 is homogeneous equation in x and y
𝟏
then 𝑴𝒙+𝑵𝒚 is an IF provided that 𝑴𝒙 + 𝑵𝒚 ≠ 𝟎.
22 Solve (𝑥 2 + 𝑦 2 )𝑑𝑥 − (𝑥 2 + 𝑥𝑦)𝑑𝑦 = 0 𝑦 𝑥
= log +𝑐
𝑥 (𝑥 − 𝑦)2
23 Solve (𝑥 2 𝑦)𝑑𝑥 − (𝑥 3 + 𝑦 3 )𝑑𝑦 = 0 𝑥3
𝑐𝑦 = 𝑒 3
3𝑦

24 Solve (𝑥 4 + 𝑦 4 ) − 𝑥𝑦 3 𝑑𝑦 = 0 4𝑥 4 𝑙𝑜𝑔𝑥 − 𝑦 4 = 𝑐𝑥 4
𝒅𝒚
Linear Differential Equations 𝒅𝒙 + 𝑷𝒚 = 𝑸
25 Solve 𝑑𝑦 + (1−2𝑥) 𝑦 = 1 𝑦 = 𝑥 2 + 𝑐𝑒 1/𝑥 𝑥 2
𝑑𝑥 𝑥2
26 2 ) 𝑑𝑦
Solve (1 − x 𝑑𝑥 + 2𝑥𝑦 = 𝑥√1 − 𝑥 2 𝑦 = √(1 − 𝑥 2 ) + 𝑐(1 − 𝑥 2 )
27 𝑑𝑦
Solve (1 − x 2 ) 𝑑𝑥 + 2𝑥𝑦 = 𝑥√1 − 𝑥 2
28 𝑑𝑦
Solve sin2x 𝑑𝑥 = 𝑦 + 𝑡𝑎𝑛𝑥 𝑦 = 𝑐 √𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑥
29 𝑑𝑦 𝑦𝑥𝑠𝑒𝑐𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐
Solve 𝑥𝑐𝑜𝑠𝑥 𝑑𝑥 + 𝑦(𝑥𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥) = 1
30 Solve (1 + 𝑥 + 𝑥𝑦 2 )𝑑𝑦 + (𝑦 + 𝑦 3 )𝑑𝑥 = 0 𝑥𝑦 + tan−1 𝑦 = 𝑐
−1 −1 −1
31 Solve (1 + 𝑦 2 )𝑑𝑥 = (𝑒 tan 𝑦 − 𝑥)𝑑𝑦 𝑥𝑒 tan 𝑦 = 𝑒 tan 𝑦 + 𝑐
32 𝑑𝑥
Solve (1 + 𝑠𝑖𝑛𝑦) 𝑑𝑦 = [2𝑦𝑐𝑜𝑠𝑦 − 𝑥(𝑠𝑒𝑐𝑦 + 𝑡𝑎𝑛𝑦)] 𝑥(1 + 𝑠𝑖𝑛𝑦) = 𝑦 2 𝑐𝑜𝑠𝑦
+ 𝑐𝑐𝑜𝑠𝑦
𝒅𝒚
The equation of the form 𝒇′ (𝒚) 𝒅𝒙 + 𝑷𝒇(𝒚) = 𝑸 where
𝑷 𝒂𝒏𝒅 𝑸 𝒂𝒓𝒆 𝒕𝒉𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒙 only can be reduce to LDE
33 𝑑𝑦
Solve 𝑑𝑥 + 𝑥𝑠𝑖𝑛2𝑦 = 𝑥 3 cos 2 𝑦 1 2
𝑡𝑎𝑛𝑦 = (𝑥 2 − 1) + 𝑐𝑒 −𝑥
2
34 𝑑𝑦
Solve 𝑡𝑎𝑛𝑦 𝑑𝑥 + 𝑡𝑎𝑛𝑥 = 𝑐𝑜𝑠𝑦𝑐𝑜𝑠 𝑥3 𝑥 𝑠𝑖𝑛2𝑥
𝑠𝑒𝑐𝑦. 𝑠𝑒𝑐𝑥 = + +𝑐
2 4
35 𝑑𝑦
Solve 𝑡𝑎𝑛𝑦 𝑑𝑥 + 𝑡𝑎𝑛𝑥 = 𝑐𝑜𝑠𝑦𝑐𝑜𝑠 3 𝑥
36 𝑑𝑦
Solve 𝑑𝑥 + (2𝑥𝑡𝑎𝑛−1 − 𝑥 3 )(1 + 𝑦 2 ) = 0 1 2 2
𝑡𝑎𝑛𝑦 = (𝑥 − 1) + 𝑐𝑒 −𝑥
2
37 𝑑𝑦
Solve 𝑑𝑥 = 1 − 2𝑥(𝑦 − 𝑥) + 𝑥 3 1 2
𝑦 − 𝑥 = (𝑥 2 − 1) + 𝑐𝑒 −𝑥
2
38 𝑑𝑦
Solve 𝑑𝑥 −
𝑡𝑎𝑛𝑦
= (1 + 𝑥)𝑒 𝑥 𝑠𝑒𝑐𝑦 𝑠𝑖𝑛𝑦
1+𝑥 = 𝑒𝑥 + 𝑐
1+𝑥
Bernoulli’s Equation
39 Solve 𝑥 𝑑𝑦 + 𝑦 = 𝑥 3 𝑦 6 1 5
𝑑𝑥 5
= 𝑥 3 + 𝑐𝑥 5
𝑦 2
40 𝑑𝑥 𝑦
Solve 𝑦 𝑑𝑦 = 𝑥 − 𝑦𝑥 2 𝑠𝑖𝑛𝑦 − = 𝑦𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑦 + 𝑐
𝑥
41 Solve 𝑦𝑑𝑥 + 𝑥(1 − 3𝑥 2 𝑦 2 )𝑑𝑦 = 0 1
+ 6𝑙𝑜𝑔𝑦 = 𝑐
𝑥 𝑦2
2

42 𝑑𝑦
Solve 𝑑𝑥 + 𝑥𝑦 = 𝑥 3 𝑦 3 1 2
= 𝑥 2 + 1 + 𝑐𝑒 𝑥
𝑦2

2|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

43 𝑥𝑦(1+𝑥𝑦 2 )𝑑𝑦 1 2
Solve =1 = 2 − 𝑦 2 + 𝑐𝑒 −𝑦 /2
𝑑𝑥
𝑥
44 𝑑𝑦
Solve 𝑑𝑥 + 𝑥(𝑥 + 𝑦) = 𝑥 3 (𝑥 + 𝑦)3 − 1 1 2
2
= 𝑥 2 + 1 + 𝑐𝑒 𝑥
(𝑥 + 𝑦)

Module 02: Higher Order Differential Equations with Constant Coefficients

Sr.
Question Solution
No.
𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
1. Solve 𝑑𝑥 3 − 6 𝑑𝑥 2 + 11 𝑑𝑥 − 6𝑦 = 0 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 3𝑥

𝑑2 𝑦 𝑑𝑦
2. Solve 2 𝑑𝑥 2 − 5 𝑑𝑥 − 12𝑦 = 0 𝑦 = 𝑐1 𝑒 3𝑥/2 + 𝑐2 𝑒 −4𝑥

𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
3. Solve 𝑑𝑥 3 − 5 𝑑𝑥 2 + 8 𝑑𝑥 − 4𝑦 = 0 𝑦 = (𝑐1 + 𝑥𝑐2 )𝑒 2𝑥 + 𝑐3 𝑒 𝑥

𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦
4. Solve 𝑑𝑥 3 − 3 𝑑𝑥 2 + 3 𝑑𝑥 − 𝑦 = 0 𝑦 = (𝑐1 + 𝑥𝑐2 + 𝑥 2 𝑐3 )𝑒 𝑥

𝑑2 𝑦 𝑑𝑦
5. . Solve 𝑑𝑥 2 + 4 𝑑𝑥 + 4𝑦 = 0 𝑦 = (𝑐1 + 𝑥𝑐2 )𝑒 −2𝑥

𝑑4 𝑦 𝑑3 𝑦 𝑑2 𝑦
6. Solve 𝑑𝑥 4 − 2 𝑑𝑥 3 + 𝑑𝑥 2 = 0 𝑦 = (𝑐1 + 𝑥𝑐2 )𝑒 𝑥 + (𝑐3 + 𝑥𝑐4 )

𝑑3 𝑦 𝑥 √3
7. Solve 𝑑𝑥 3 + 𝑦 = 0 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑒 2 (𝑐2 cos √3 /2 𝑥 + 𝑐3 sin 𝑥)
2
𝑑4 𝑦 𝑑2 𝑦
8. Solve 𝑑𝑥 4 + 6 𝑑𝑥 2 + 9𝑦 = 0 𝑦 = (𝑐1 + 𝑥𝑐2 ) cos 𝑥 + (𝑐3 + 𝑐4 𝑥)𝑠𝑖𝑛𝑥

𝑑4 𝑦 √3 √3
9. Solve 𝑑𝑥 4 + 4𝑦 = 0 𝑦 = 𝑒 𝑥/2 {(𝑐1 + 𝑥𝑐2 )𝑐𝑜𝑠 𝑥 + (𝑐3 + 𝑥𝑐4 )𝑠𝑖𝑛 𝑥
2 2

10. Solve (𝐷 2 − 2𝐷 − 4)) = 0 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠ℎ√5 𝑥 + 𝑐2 sinh √5 𝑥)

−𝑥
𝑒 3𝑥
11 Solve
𝑑3 𝑦
−3
𝑑2 𝑦
+ 4𝑦 = 𝑒 3𝑥 𝑦 = 𝑐1 𝑒 + (𝑐2 + 𝑥𝑐3 )𝑒 2𝑥 +
𝑑𝑥 3 𝑑𝑥 2 4

𝑥 3𝑥 4
12 Solve (𝐷 3 − 2𝐷 2 − 5𝐷 + 6)𝑦 = 𝑒 3𝑥 + 8 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 3𝑥 + 𝑒 +
10 3
𝑑3 𝑦 𝑑𝑦 𝑥 1
13 Solve 𝑑𝑥 3 − 4 𝑑𝑥 = 2 𝑐𝑜𝑠ℎ2 2𝑥 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑐3 𝑒 −2𝑥 − + 𝑠𝑖𝑛ℎ4𝑥
4 48
Find the particular integral of 1
14
(𝐷 2 − 4𝐷 + 4)𝑦 = 𝑒 𝑥 + 𝑐𝑜𝑠2𝑥 𝑦 = 𝑒 𝑥 − 𝑠𝑖𝑛2𝑥
8
𝑥
15 Solve (𝐷 2 + 4)𝑦 = 𝑐𝑜𝑠2𝑥 𝑦 = 𝑐1 𝑐𝑜𝑠2𝑥 + 𝑐3 𝑠𝑖𝑛2𝑥 + 𝑠𝑖𝑛2𝑥
4
1
16 Solve (𝐷 2 − 5𝐷 + 6)𝑦 = 𝑠𝑖𝑛3𝑥 𝑦 = 𝑐1 𝑒 2𝑥 + 𝑐2 𝑒 3𝑥 + (5𝑐𝑜𝑠3𝑥 − 𝑠𝑖𝑛3𝑥)
78

3|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

𝑑3 𝑦 𝑑2 𝑦 𝑑𝑦 𝑥 𝑥
17 Solve 𝑑𝑥 3 − 3 𝑑𝑥 2 + 9 𝑑𝑥 − 27𝑦 = 𝑐𝑜𝑠3𝑥 𝑦 = 𝑐1 𝑒 3𝑥 + (𝑐2 𝑐𝑜𝑠3𝑥 + 𝑐3 𝑠𝑖𝑛3𝑥) − 𝑐𝑜𝑠3𝑥 − 𝑠𝑖𝑛3𝑥
36 36

𝑥 1
𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 + 𝑐3 𝑐𝑜𝑠𝑥 + 𝑐4 𝑠𝑖𝑛𝑥 + 𝑒 𝑥 + 𝑐𝑜𝑠4𝑥
18 Solve (𝐷 4 − 1)𝑦 = 𝑒 𝑥 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠3𝑥 4 510
1
+ 𝑐𝑜𝑠2𝑥
30
1 𝑥2
19 Solve (𝐷 4 + 8𝐷 2 + 16)𝑦 = 𝑠𝑖𝑛2 𝑥 𝑦 = (𝑐1 + 𝑥𝑐2 )(𝑐3 𝑐𝑜𝑠2𝑥 + 𝑐4 𝑠𝑖𝑛2𝑥) + + 𝑐𝑜𝑠2𝑥
32 64
1 3
20 Solve (𝐷 3 − 3𝐷 + 2)𝑦 = 𝑥 𝑦 = (𝑐1 + 𝑥𝑐2 )𝑒 𝑥 + 𝑐3 𝑒 −2𝑥 + [𝑥 + ]
2 2
𝑑3 𝑦 𝑑𝑦 1
21 Solve 𝑑𝑥 3 − 2 𝑑𝑥 + 4𝑦 = 3𝑥 2 − 5𝑥 + 2 𝑦 = 𝑐1 𝑒 −2𝑥 + 𝑒 𝑥 (𝑐2 𝑐𝑜𝑠𝑥 + 𝑐3 𝑠𝑖𝑛𝑥) + [3𝑥 2 − 2𝑥 + 1]
4

𝑥 3 5𝑥 2
22 SSolve (𝐷 3 − 2𝐷 2 + 𝐷)𝑦 = 𝑥 2 + 𝑥 𝑦 = 𝑐1 + (𝑐2 + 𝑥𝑐3 )𝑒 𝑥 + + + 8𝑥
3 2
1 𝑥 3 𝑥 2 25𝑥
23 Solve (𝐷 3 − 𝐷 2 − 6𝐷)𝑦 = 𝑥 2 + 1 𝑦 = 𝑐1 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 3𝑥 − [ − + ]
6 3 6 1
𝑥3
24 Solve (𝐷 2 − 3𝐷 + 2)𝑦 = 𝑥 2 𝑒 2𝑥 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 + 𝑒 2𝑥 ( − 𝑥 2 + 2𝑥)
3
𝑒 3𝑥 2 3
25 Solve (𝐷 2 − 4𝐷 + 3)𝑦 = 2𝑥𝑒 3𝑥 + 3𝑒 𝑥 𝑐𝑜𝑠2𝑥 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 3𝑥 + (𝑥 − 𝑥) − 𝑒 𝑥 (𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥)
2 8
𝑒 3𝑥 2 12𝑥 50 𝑒𝑥
𝑑2 𝑦 𝑦 = 𝑐1 𝑐𝑜𝑠√2𝑥 + 𝑐2 𝑠𝑖𝑛√2𝑥 + {𝑥 − + }+
26 Solve 𝑑𝑥 2 + 2𝑦 = 𝑥 2 𝑒 3𝑥 + 𝑒 𝑥 − 𝑐𝑜𝑠2𝑥 11 11 121 3
1
+ 𝑐𝑜𝑠2𝑥
2
𝑒 3𝑥 2 12 50
𝑦 = 𝑐1 𝑐𝑜𝑠√2𝑥 + 𝑐2 𝑠𝑖𝑛√2𝑥 + {𝑥 − 𝑥 + }
27 Solve (𝐷 2 + 2)𝑦 = 𝑒 𝑥 𝑐𝑜𝑠𝑥 + 𝑥 2 𝑒 3𝑥 11 11 121
𝑒𝑥
+ (𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥)
4
1 𝑥
𝑦 = 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑐3 𝑒 3𝑥 − 𝑒 (𝑠𝑖𝑛𝑥 + 3𝑐𝑜𝑠𝑥)
28 Solve (𝐷 3 − 7𝐷 − 6)𝑦 = 𝑐𝑜𝑠ℎ𝑥𝑐𝑜𝑠𝑥 100
1 −𝑥
+ 𝑒 (3𝑐𝑜𝑠𝑥 − 5𝑠𝑖𝑛𝑥)
68
𝑥 2 3𝑥 𝑥 3 3𝑥
29 Solve (𝐷 2 − 6𝐷 + 9)𝑦 = 𝑒 3𝑥 (1 + 𝑥) 𝑦 = (𝑐1 + 𝑥𝑐2 )𝑒 3𝑥 + 𝑒 + 𝑒
2 6

Apply the method of variation of parameters to 1 𝑥

30 𝑦 = 𝑐1 𝑐𝑜𝑠𝑎𝑥 + 𝑐2 𝑠𝑖𝑛𝑎𝑥 + 𝑙𝑜𝑔𝑐𝑜𝑠𝑎𝑥. 𝑐𝑜𝑠𝑎𝑥 + 𝑠𝑖𝑛𝑎𝑥
solve (𝐷 2 + 𝑎2 )𝑦 = 𝑠𝑒𝑐𝑎 𝑥 𝑎2 𝑎
Apply the method of variation of parameters to 𝑥
31 d2 𝑦 𝑑𝑦 𝑥 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −2𝑥 + 𝑒 −2𝑥 𝑒 𝑒
solve 𝑑𝑥 2 + 3 𝑑𝑥 + 2𝑦 = 𝑒 𝑒
Apply the method of variation of parameters to 𝑦 = 𝑐1 𝑐𝑜𝑠𝑥 + 𝑐2 𝑠𝑖𝑛𝑥 + (𝑥 − 𝑡𝑎𝑛𝑥)𝑐𝑜𝑠𝑥
32 d2 𝑦
solve + 𝑦 = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 + 𝑠𝑖𝑛𝑥𝑙𝑜𝑔𝑠𝑒𝑐𝑥
𝑑𝑥 2
Apply the method of variation of parameters to
33 𝑦 = 𝑐1 𝑒 2𝑥 + 𝑐2 𝑥𝑒 2𝑥 + 𝑒 2𝑥 𝑙𝑜𝑔𝑠𝑒𝑐𝑥
solve (𝐷 2 − 4𝐷 + 4)𝑦 = 𝑒 2𝑥 sec 2 𝑥
Apply the method of variation of parameters to
34 𝑦 = 𝑐1 𝑒 3𝑥 + 𝑐2 𝑥𝑒 3𝑥 − 𝑒 3𝑥 (𝑙𝑜𝑔𝑥 + 1)
solve (𝐷 2 − 6𝐷 + 9)𝑦 = 𝑒 3𝑥 /𝑥 2
Apply the method of variation of parameters to 𝑦 = 𝑒 𝑥 (𝑐1 𝑐𝑜𝑠𝑥 + 𝑐2 𝑠𝑖𝑛𝑥) − 𝑒 𝑥 𝑐𝑜𝑠𝑥. log (𝑠𝑒𝑐𝑥
35
solve (𝐷 2 − 2𝐷 + 2)𝑦 = 𝑒 𝑥 𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑥)

4|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

Apply the method of variation of parameters to 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 −𝑥 − 1 + 𝑒 𝑥 . log(1 + 𝑒 −𝑥 )

36 2
solve (𝐷 2 − 1)𝑦 = − 𝑒 −𝑥 . log (1 + 𝑒 𝑥 )
1+𝑒 𝑥
Apply the method of variation of parameters to 𝑦 = 𝑐1 𝑒 𝑥 + 𝑐2 𝑒 2𝑥 − 𝑒 𝑥 + (𝑒 𝑥
37 𝑒𝑥
solve (𝐷 2 − 3𝐷 + 2)𝑦 = 𝑥 + 𝑒 2𝑥 ). log(1 + 𝑒 −𝑥 )
1+𝑒
Apply the method of variation of parameters to
38 𝑦 = 𝑐1 𝑒 −𝑥 + 𝑐2 𝑒 2𝑥 − 𝑒 −2𝑥 sin 𝑒 𝑥
solve (𝐷 2 + 3𝐷 + 2)𝑦 = 𝑠𝑖𝑛 𝑒 𝑥

Module 03: Beta and Gamma Function, Differentiation under Integral

sign, and Rectification
Beta Gamma Functions
∞ 𝟒 1 𝟏
1. Evaluate ∫𝟎 𝒆−𝒙 . 𝒅𝒙 𝚪( )
4 𝟒
3
2. Evaluate
∞
∫0 √𝑥 𝑒 − √𝑥 . 𝑑𝑥 315
√𝜋
16
∞ 3
3. Evaluate ∫0 𝑥 1/4 𝑒 −√𝑥 𝑑𝑥 √𝜋
2
∞ 8
1 𝟑
4. Evaluate ∫0 𝑥 2 𝑒 −𝑥 𝑑𝑥 𝚪( )
8 𝟖
∞ 2
5. Evaluate ∫0 (𝑥 4 + 4)𝑒 −2𝑥 𝑑𝑥 67 𝜋
√
32 2
∞ 8 ∞ 4 𝜋
6. Show that a) ∫0 𝑥𝑒 −𝑥 𝑑𝑥. ∫0 𝑥 2 𝑒 −𝑥 𝑑𝑥 =
16 √2
2 2
∞ 𝑒 −𝑥 ∞ 6 𝜋 ∞ 2 ∞ 𝑒 −𝑦 𝑑𝑥 𝜋
b) ∫0 𝑥 𝑑𝑥. ∫0 𝑥 4 𝑒 −𝑦 𝑑𝑥 = 9
, c) ∫0 √𝑦. 𝑒 −𝑦 𝑑𝑦. ∫0 =2
√ √𝑦 √2
7. 1 1 Γ(𝑛 + 1)
Evaluate ∫0 𝑥 𝑚 (log 𝑥)𝑛 𝑑𝑥
(𝑚 + 1)𝑛+1
8. 1
Evaluate ∫0 (𝑥𝑙𝑜𝑔𝑥)3 𝑑𝑥 3
−
128
9. ∞ 𝑥4 4!
Evaluate ∫0 𝑑𝑥
4𝑥 (log 4)5
∞ 2 1
10. Evaluate ∫0 𝑥 2 . 7−4𝑥 𝑑𝑥 √𝜋
32(log 7)3/2
11. ∞ 𝚪(𝑚) 𝑚𝜋
Show that ∫0 𝑥 𝑚−1 cos 𝑎𝑥 𝑑𝑥 = 𝑎𝑚
cos (2
)
∞ Γ (𝑛 + 1) 𝑛𝜋
12. Evaluate ∫0 cos(𝑎𝑥 1/𝑛 ) 𝑑𝑥
cos ( )
𝑎𝑛 2
13. 2 16/15
Evaluate ∫0 𝑥 2 (2 − 𝑥)3 𝑑𝑥
8
14. Evaluate ∫0 𝑥 4 (8 − 𝑥)−1/3 𝑑𝑥 85 2
. 𝐵 (5, )
3 3
15. 1
Evaluate ∫0 𝑥 6 (1 − 𝑥 2 )1/2 𝑑𝑥 1 7 3
𝐵( , )
2 2 2
16. 1 5 3
Evaluate ∫0 √[√𝑥 − 𝑥]. 𝑑𝑥 2𝐵 ( , )
2 2
17. 2𝑎
Evaluate ∫0 𝑥 𝑚 √2𝑎𝑥 − 𝑥 2 𝑑𝑥 3 3
(2𝑎)𝑚+2 𝐵 (𝑚 + , )
2 2

5|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

18. 2𝑎 5 4
Evaluate ∫0 𝑥 2 √2𝑎𝑥 − 𝑥 2 𝑑𝑥 𝑎 𝜋
8
19. 3 𝑥 3\2 1 𝑑𝑥 432
Prove that a) ∫0 𝑑𝑥. ∫0 = 𝜋
√3−𝑥 √1−𝑥 1/4 35

1 1\2 𝜋
b) ∫0 √1 − √𝑥 𝑑𝑥. ∫0 √2𝑦 − 4𝑦 2 𝑑𝑦 = 30

20. 𝟐𝒂 𝒙𝟗/𝟐 11 1
Evaluate ∫𝟎 𝒅𝒙 (2𝑎)5. 𝐵 ( , )
√𝟐𝒂−𝒙 2 2
21. 1 𝑥7 1/3
Evaluate ∫0 𝑑𝑥
√1−𝑥 4

22. ∞ 𝒅𝒙 16/35
Evaluate ∫𝟎 (𝟏+𝒙𝟐 )𝟗/𝟐
23. ∞ 𝒙𝟐 8/45
Evaluate ∫𝟎 (𝟏+𝒙𝟔 )𝟕/𝟐 𝒅𝒙

24. ∞ 𝒙𝟑 𝟓
Evaluate ∫𝟎 (𝟏+𝒙𝟖 )𝟒
dx 𝝅
𝟏𝟐𝟖
25. 𝜋/6 16/315
Evaluate ∫0 𝑐𝑜𝑠 3 3𝜃 𝑠𝑖𝑛2 6𝜃 𝑑𝜃
2𝑎
26. Evaluate ∫0 𝑥√2𝑎𝑥 − 𝑥 2 𝑑𝑥 𝝅𝒂𝟑
𝟐
27. 1/2 1/120
Evaluate ∫0 𝑥 3 √1 − 4𝑥 2 . 𝑑𝑥
28. ∞ 𝑥2 2/15
Evaluate ∫0 (1+𝑥 2 )7/2
𝑑𝑥
29. 𝑎 𝑑𝑥 𝜋 𝜋
Show that ∫0 (𝑎𝑛 −𝑥𝑛 )1/𝑛 = 𝑐𝑜𝑠𝑒𝑐 ( )
𝑛 𝑛

30. ∞ 𝑥 5 (1+𝑥 4 ) B(6,10)

Express ∫0 𝑑𝑥
(1+𝑥)16

31. ∞ 𝑑𝑥 1 𝑛 𝑛 ∞ 16/35
Prove that ∫0 = 𝐵 ( , ) and hence evaluate ∫0 𝑠𝑒𝑐ℎ 8 𝑥 𝑑𝑥
(𝑒 𝑥 +𝑒 −𝑥 )𝑛 4 2 2

Differentiation Under Integral sign

32 1 𝑥 𝛼 −1 1 𝑥 7 −1 𝐥𝐨𝐠 𝟖
Prove that ∫0 log 𝑥
𝑑𝑥 = log(1 + 𝛼) , 𝛼 ≥ 0.Hence , evaluate ∫0 log 𝑥
𝑑𝑥
33 ∞ 𝑒 −𝑥 𝐥𝐨𝐠 𝟖
Evaluate ∫0 (1 − 𝑒 −𝑎𝑥 )𝑑𝑥 , 𝑎 > −1 .Hence,
𝑥
∞ 𝑒 −𝑥
evaluate ∫0 (1 − 𝑒 −7𝑥 )𝑑𝑥
𝑥

34 ∞ sin 𝑚𝑥 𝑚 𝝅/𝟒
Prove that ∫0 𝑒 −𝑎𝑥 . 𝑥
𝑑𝑥 = tan−1 𝑎 . ( 𝑎 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟) Hence ,
∞ sin 𝑥
evaluate ∫0 𝑒 −𝑥 𝑥 . 𝑑𝑥
35 Show that ∫0
𝜋 log(1+𝑎 cos 𝑥)
𝑑𝑥 = 𝜋𝑠𝑖𝑛−1 𝑎 , 0 ≤ 𝑎 ≤ 1 .Hence 𝝅𝟐
cos 𝑥
𝟐
𝜋 log(1+cos 𝑥)
evaluate∫0 cos 𝑥
𝑑𝑥

36 𝜋 log(1+sin 𝛼 cos 𝑥)
Show that ∫0 𝑑𝑥 = 𝜋𝛼
cos 𝑥

6|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology
𝜋
37 log(1+cos 𝛼 cos 𝑥) 𝜋2 𝛼2
Show that ∫02 𝑑𝑥 = −
cos 𝑥 8 2

38 ∞ 1−cos 𝑚𝑥 1
Show that ∫0 𝑥
. 𝑒 −𝑥 𝑑𝑥 = 2
log(𝑚2 + 1). Hence, deduce that
∞ 1−cos 𝑥
∫0 . 𝑒 −𝑥 𝑑𝑥 = log √2
𝑥
𝜋
39 log(1+𝑎𝑠𝑖𝑛2 𝑥)
Prove that ∫0 2 𝑑𝑥 = 𝜋[√𝑎 + 1 − 1], 𝑎 > −1
𝑠𝑖𝑛2 𝑥

Rectification
1 Find the total length of the curve 𝑥 2/3 + 𝑦 2/3 = 𝑎2/3 𝟔𝒂

2 Show that the length of the arc of the curve 𝑎𝑦 2 = 𝑥 3 from the origin to the
8𝑎 9𝑏
point whose abscissa is b is 27 [(1 + 4𝑎) − 13/2 ]
3 Show that if s is the arc of the curve 9𝑦 2 = 𝑥(3 − 𝑥)2 measured from the origin 𝟑𝒔𝟐
to the point P(x,y) then 3𝑠 2 = 3𝑦 2 + 4𝑥 2
4 Find the perimeter of the loop of the curve 4𝑎√3
9𝑎𝑦 2 = (𝑥 − 2𝑎)(𝑥 − 5𝑎)2
5 Find the length of the loop of the curve 3𝑎𝑦 2 = 𝑥(𝑥 − 𝑎)2 𝟒
𝒂
√𝟑
6 Find the total length of the loop of the curve 4√3
9𝑦 2 = (𝑥 + 7)(𝑥 + 4)2
7 Find the perimeter of the cardioid 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃) and prove that the line 𝜃 = 8a
2𝜋/3 bisects the upper half of the cardioid
8 Find the perimeter of the 𝑐𝑎𝑟𝑑𝑖𝑜𝑖𝑑 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃) 8a
9 Find the length of the cardioid 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃), 𝑙𝑦𝑖𝑛𝑔 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑟 = 4𝑎√3
𝑎𝑐𝑜𝑠𝜃
10 Find the length of the cardioid 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃), 𝑙𝑦𝑖𝑛𝑔 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑟 + 4𝑎√3
𝑎𝑐𝑜𝑠𝜃 = 0
11 Find the length of the upper arc of one loop of Lemniscate 𝑟 2 = 𝑎2 cos 2𝜃 𝑎 (Γ 1/4)2
.
4√2 √𝜋

Module 04: Multiple Integration I [Double Integration]

4.1 Evaluation of Double Integral
1 𝑥
Evaluate ∫0 ∫𝑥 2 𝑥𝑦 (𝑥 + 𝑦)𝑑𝑦 𝑑𝑥 3
1. [ ]
56
1 √1+𝑥 2 𝑑𝑥 𝑑𝑦 𝜋
2. Evaluate ∫0 ∫0 [ log (1 + √2)]
1+𝑥 2 +𝑦 2 4

1 𝑥2 𝑦⁄ 1
3. Evaluate ∫0 ∫0 𝑒 𝑥 𝑑𝑦 𝑑𝑥 [ ]
2
3
1 𝑥 [ ]
4. Evaluate ∫0 ∫𝑥 2 𝑥𝑦 (𝑥 2 + 𝑦 2 )𝑑𝑦 𝑑𝑥 80

7|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

1 𝑥 1
5. Evaluate ∫0 ∫0 𝑒 𝑥+𝑦 ⋅ 𝑑𝑥 𝑑𝑦 [𝑒 − 1]2
2
1 𝑦 2
6. Evaluate ∫0 ∫0 𝑥𝑦𝑒 −𝑥 ⋅ 𝑑𝑥 𝑑𝑦 1/4e

1 𝑥
7. Evaluate ∫0 ∫0 (𝑥 2 + 𝑦 2 )𝑥 ⋅ 𝑑𝑥 𝑑𝑦 4/15

5 2+𝑥
8. Evaluate ∫0 ∫2−𝑥 𝑑𝑥 𝑑𝑦 25

𝜋/4 √𝑐𝑜𝑠2𝜃 𝑟 1
9. Evaluate ∫0 ∫0 (1+𝑟 2 )2
⋅ 𝑑𝑟 𝑑𝜃 (𝜋 − 2)
8
𝜋/2 𝑎𝑐𝑜𝑠𝜃 2 2𝑎3
10. Evaluate ∫0 ∫0 𝑟 ⋅ 𝑑𝑟 𝑑𝜃
9
𝜋 𝑎(1+𝑐𝑜𝑠𝜃) 3𝜋𝑎2
11. Evaluate ∫0 ∫0 𝑟 ⋅ 𝑑𝑟 𝑑𝜃
4
4.2 Evaluation of Double Integrals by changing the order of integration
2 4−𝑥
12. Change the order of integration and evaluate ∫0 ∫𝑥2 𝑥𝑦 𝑑𝑦 𝑑𝑥 6
2

Change the order of integration and evaluate

13. 2 2+√4−𝑦2 2𝜋
∫0 ∫2−√4−𝑦 2 𝑑𝑥 𝑑𝑦
Change the order of integration and evaluate √2 − 1
14. 1 √2−𝑥 2 𝑥 𝑑𝑥 𝑑𝑦
∫0 ∫𝑥 𝑑𝑥 𝑑𝑦 √2
√𝑥 2 +𝑦 2
Change the order of integration and evaluate 2𝜋
15. 2 2 𝑥2 [ ]
∫0 ∫√2𝑦 √𝑥 4 𝑑𝑥 𝑑𝑦 3
−4𝑦 2

1 4 2 1 16
16. Change the order of integration and evaluate ∫0 ∫4𝑦 𝑒 𝑥 𝑑𝑥 𝑑𝑦 [𝑒 − 1]
8
5 2+𝑥
17. Change the order of integration and evaluate ∫0 ∫2−𝑥 𝑑𝑥 𝑑𝑦 25

1 2−𝑥 𝑥
18. Change the order of integration and evaluate ∫0 ∫𝑥 𝑑𝑥 𝑑𝑦 log(4/e)
𝑦

Change the order of integration and evaluate ∫ ∫𝑅 𝑥 2 𝑑𝑥 𝑑𝑦,

7𝑎4
19. where R is the region in the first quadrant bounded by 𝑥𝑦 = 𝑎2 ,
4
𝑥 = 2𝑎, 𝑦 = 0, 𝑎𝑛𝑑 𝑦 = 𝑥
4.3 Evaluation of Integrals over the given region (Cartesian and Polar)
Evaluate ∬ 𝑥𝑦 (𝑥 − 1)𝑑𝑥 𝑑𝑦 over the region bounded by
[8(3 − 2 𝑙𝑜𝑔2)]
21. 𝑥𝑦 = 4, 𝑦 = 0, 𝑥 = 1 𝑎𝑛𝑑 𝑥 = 4

8|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

𝑎4
22. Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦
3
1
23. Evaluate ∫ ∫𝑅 4 2 𝑑𝑥 𝑑𝑦, 𝑤ℎ𝑒𝑟𝑒 𝑅 𝑖𝑠 𝑟𝑒𝑔𝑖𝑜𝑛 𝑥 ≥ 1, 𝑦 ≥ 𝑥 2 𝜋/4
𝑥 +𝑦

Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 1
24. −
𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎𝑠 𝑦 = 𝑥 2 𝑎𝑛𝑑 𝑦 2 = −𝑥 12
Evaluate ∫ ∫ 𝑥 2 𝑑𝑥 𝑑𝑦 over the region bounded by 𝑥𝑦 = 16, 𝑦 =
25. 448
𝑥, 𝑦 = 0, 𝑎𝑛𝑑 𝑥 = 8

Evaluate ∬ 𝑥 (𝑥 − 𝑦)𝑑𝑥 𝑑𝑦 over the triangle region with vertices

26. -1
(0,0), (1,2), (0,4)

Evaluate ∫ ∫ (𝑥 2 + 𝑦 2 ) 𝑑𝑥 𝑑𝑦 over the triangle whose vertices are

27. 7/6
(0,1), (1,1), (1,2)

Evaluate ∫ ∫ √𝑥𝑦 − 𝑦 2 𝑑𝑥 𝑑𝑦, over the triangle whose vertices are

28. 6
(0,0), (10,1), (1,1)

Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦

29. 7/12
𝑥 2 + 𝑦 2 = 2𝑥, 𝑦 2 = 2𝑥, 𝑦 = 𝑥

Evaluate ∫ ∫ 𝑟 3 𝑑𝑟 𝑑𝜃 over the circles. 45

30. 𝜋
𝑟 = 2𝑠𝑖𝑛𝜃 𝑎𝑛𝑑 𝑟 = 4𝑠𝑖𝑛𝜃 2
𝑟2
Evaluate ∫ ∫ 𝑟𝑒
− 2
𝑎 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃 𝑑𝑟 𝑑𝜃 over the upper half of the circle 𝑎2
31. [3 + 𝑒 −4 ]
𝑟 = 2𝑎𝑐𝑜𝑠𝜃 16
𝑟 𝑑𝑟 𝑑𝜃
Evaluate ∫ ∫ 2 2 , over one loop of lemniscate. 𝑎
32. √𝑟 +𝑎 [4 − 𝜋]
𝑟 2 = 𝑎2 𝑐𝑜𝑠2𝜃 2
𝑑𝑥 𝑑𝑦
Change to polar coordinates and evaluate ∬ (1+𝑥2 +𝑦2 )2 over one 𝜋−2
33. [ ]
loop of the lemniscates (𝑥 2 + 𝑦 2 )2 = 𝑥 2 − 𝑦 2 4
5.1 Evaluation of Double Integrals by changing to polar coordinates.
Change to polar coordinates and evaluate ∬ 𝑦 2 𝑑𝑥 𝑑𝑦 over the 15 𝜋𝑎4
34. area outside 𝑥 2 + 𝑦 2 − 𝑎𝑥 = 0 𝑎𝑛𝑑 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 [ ]
2 2 16
𝑥 + 𝑦 − 2𝑎𝑥 = 0
1 𝑥
Change to polar coordinates and evaluate ∫0 ∫0 (𝑥 + 𝑦) 𝑑𝑦 𝑑𝑥 1
35. [2]
𝑎 𝑎
Change to polar coordinates and evaluate ∫0 ∫𝑦 𝑥 𝑑𝑦 𝑑𝑥
36. 𝑎3 /3

Change to polar coordinates and evaluate 𝜋 5

37. 4𝑎 𝑦 𝑥 2 −𝑦 2 8𝑎2 [ − ]
∫0 ∫𝑦 2/4𝑎 [𝑥 2 +𝑦 2] 𝑑𝑥 𝑑𝑦 2 3
Change to polar coordinates and evaluate 𝜋
38. ∞ ∞ 2 2
∫0 ∫0 𝑒 −(𝑥 +𝑦 ) 𝑑𝑦 𝑑𝑥 4

9|Page First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

Use polar coordinates to evaluate ∬ √𝑎2 − 𝑥 2 − 𝑦 2 𝑑𝑥 𝑑𝑦 𝑎3

39. [3𝜋 − 4]
over the upper half of the circle 𝑥 2 + 𝑦 2 = 𝑎𝑥 18
1
Use polar coordinates to evaluate ∫ ∫ 𝑑𝑥 𝑑𝑦, over the 𝜋
40. √𝑥𝑦
2 2
region bounded by 𝑥 + 𝑦 = 𝑥, 𝑎𝑛𝑑 𝑦 ≥ 0 √2
Area by Double Integration

44. Find by double integration the area enclosed by 𝑦 2 = 𝑥 3 𝑎𝑛𝑑 𝑦 = 𝑥 1/10

Find by double integration the area enclosed by the curve 1 4

45.
9𝑥𝑦 = 4 𝑎𝑛𝑑 𝑎 𝑙𝑖𝑛𝑒 2𝑥 + 𝑦 = 2 − 𝑙𝑜𝑔2
3 9
Find by double integration the area between the curve
46. 4𝜋𝑎2
𝑦 2 𝑥 = 4𝑎2 (2𝑎 − 𝑥) 𝑎𝑛𝑑 𝑖𝑡𝑠 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
Find by double integration the area between the curve
47. 3𝜋𝑎2
𝑦 2 (2𝑎 − 𝑥) = 𝑥 3 𝑎𝑛𝑑 𝑖𝑡𝑠 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
3𝜋𝑎2
48. Find by the double integration the area of cardioide 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃)
2
Find by the double integration the total area of lemnicate
49. 𝑎2
(𝑥 2 + 𝑦 2 )2 = 𝑎2 (𝑥 2 − 𝑦 2 )
Find by the double integration the area inside the circle 𝑟 = 𝑎𝑠𝑖𝑛𝜃 and 𝑎2
50. [4 − 𝜋]
outside the cardioide. 𝑟 = 𝑎(1 − 𝑐𝑜𝑠𝜃) 4
Find by the double integration the area between the circles 𝑟 = 2𝑎𝑠𝑖𝑛𝜃 and
51. (𝑏 2 − 𝑎2 )𝜋
𝑟 = 2𝑏𝑠𝑖𝑛𝜃, (𝑏 > 𝑎)
Find by the double integration the area outside the circle 𝑟 = 𝑎 and inside 𝑎2
52. [8 + 𝜋]
the cardioide. 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) 4
Find by the double integration the area common to the circles 𝑟 = 𝑎 and 2𝜋𝑎2 √3 2
53. − 𝑎
𝑟 = 2𝑎𝑐𝑜𝑠𝜃. 3 2

Module 05: Multiple Integration II [Triple Integration]

Evaluation of Triple Integral
1 1 1−𝑥
1. Evaluate ∫0 ∫𝑦 2 ∫0 𝑥 𝑑𝑧 𝑑𝑦 𝑑𝑥 4/35
2 2 𝑦𝑧
2. Evaluate ∫0 ∫1 ∫0 𝑥𝑦𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥 15/2

2 𝑥 2𝑥+2𝑦 𝑒 12 𝑒 6 𝑒 4 4
3. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥 [
18
− − + 𝑒2 − ]
9 2 9
𝑙𝑜𝑔2 𝑥 𝑥+𝑦 5
4. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥 [ ]
8
𝑒 𝑙𝑜𝑔𝑦 𝑒𝑥 1
5. Evaluate ∫1 ∫1 ∫1 𝑙𝑜𝑔𝑧 𝑑𝑧 𝑑𝑥 𝑑𝑦 [ (𝑒 2 − 8𝑒 + 13)]
2

10 | P a g e First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

1 1−𝑥 1−𝑥−𝑦
1 5
1 [ (𝑙𝑜𝑔2 − )]
6. Evaluate ∫0 ∫0 ∫0 (𝑥+𝑦+𝑧+1)3
𝑑𝑧 𝑑𝑦 𝑑𝑥 2 8

𝑎 𝑎−𝑥 𝑎−𝑥−𝑦 𝑎5
7. Evaluate ∫0 ∫0 ∫0 𝑥 2 𝑑𝑧 𝑑𝑦 𝑑𝑥
60
1 𝑧 𝑥+𝑧
8. Evaluate ∫−1 ∫0 ∫𝑥−𝑧 (𝑥 + 𝑦 + 𝑧) 𝑑𝑧 𝑑𝑦 𝑑𝑥 0

1 1−𝑥 𝑥+𝑦
9. Evaluate ∫0 ∫0 ∫0 𝑒 𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥 1/2
𝑎 𝑥 𝑥+𝑦 𝑎3
10. Evaluate ∫0 ∫0 ∫0√ 𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
4
When the region of integration is bounded by planes

Evaluate the integral ∭ 𝑥𝑦𝑧 2 𝑑𝑣 over the region bounded by the 27

11.
planes 𝑥 = 0, 𝑥 = 1, 𝑦 = −1, 𝑦 = 2, 𝑧 = 0, 𝑧 = 3. 4
Evaluate the integral ∭ 𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the volume of the 𝑎𝑏𝑐 2
12. 𝑥 𝑦 𝑧
tetrahedron bounded by the 𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑎 + 𝑏 + 𝑐 = 1. 24

Evaluate the integral ∭(𝑥 + 𝑦 + 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the tetrahedron 1

13.
bounded by the 𝑥 = 0, 𝑦 = 0, 𝑧 = 0 𝑎𝑛𝑑 𝑥 + 𝑦 + 𝑧 = 1. 8

When the region of integration is not bounded by planes, but by sphere, ellipsoid etc.

Evaluate ∭ 𝑥𝑦𝑧(𝑥 2 + 𝑦 2 + 𝑧 2 ) 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the first octant 𝑎8

14. [ ]
of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2 . 64
Evaluate ∭ 𝑥𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the positive octant of the 𝑎6
15.
sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2 48
Evaluate ∭(𝑥 2 + 𝑦 2 + 𝑧 2 ) 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the first octant of 𝑎5
16. 𝜋⋅
the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2 10
𝑑𝑥 𝑑𝑦 𝑑𝑧
Evaluate ∭ (𝑥 2 +𝑦 2+𝑧 2) throughout the volume of the sphere
17. [4𝜋𝑎]
2 2 2 2
𝑥 +𝑦 +𝑧 = 𝑎 .
𝑧 2 𝑑𝑥 𝑑𝑦 𝑑𝑧
Evaluate ∭ (𝑥 2 +𝑦 2+𝑧 2) throughout the volume of the sphere 𝜋√2
18. 8⋅
2 2 2 9
𝑥 + 𝑦 + 𝑧 = 2.
𝑑𝑥 𝑑𝑦 𝑑𝑧
Evaluate ∭ (𝑥 2+𝑦 2+𝑧 2)3/2 throughout the volume bounded by 𝑏
19. 4𝜋 log ( )
2 2 2 2
sphere 𝑥 + 𝑦 + 𝑧 = 𝑎 𝑎𝑛𝑑 𝑥 + 𝑦 + 𝑧 = 𝑏 , (𝑏 > 𝑎) 2 2 2 2 𝑎
When the region of integration is bounded by cone or a cylinder or a paraboloid

Evaluate ∭ √𝑥 2 + 𝑦 2 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the volume bounded by the 𝜋

21. [ ]
right circular cone 𝑥 2 + 𝑦 2 = 𝑧 2 , 𝑧 > 0 and the planes 𝑧 = 0, 𝑧 = 1. 6
Evaluate ∭ 𝑧 2 𝑑𝑥 𝑑𝑦 𝑑𝑧 over the volume common to the 2 𝜋𝑎5
22. [ ]
sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2 and the cylinder 𝑥 2 + 𝑦 2 = 𝑎𝑥. 15
Volume of solids
Find the volume of tetrahedron bounded by the planes 𝑎3
1
𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 + 𝑦 + 𝑧 = 𝑎 6

11 | P a g e First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

Find the volume bounded by the paraboloid 𝑥 2 + 𝑦 2 = 𝑎𝑧 and 𝜋𝑎3

2
the cylinder 𝑥 2 + 𝑦 2 = 𝑎2 2
Find the volume bounded by the cylinder 𝑥 2 + 𝑦 2 = 𝑎2 and the
3 𝜋𝑎2 𝑏
planes 𝑧 = 0, 𝑦 + 𝑧 = 𝑏
Find the volume bounded by the cone 𝑧 2 = 𝑥 2 + 𝑦 2 and
4 𝜋/6
paraboloid 𝑥 2 + 𝑦 2 = 𝑧

Module 06: Numerical solution of ordinary differential equations of first order

and first degree and Numerical Integration
Euler’s Method
Using Euler’s method find approximate value of 𝑦 𝑎𝑡 𝑥 = 1 in five steps
1. 𝑑𝑦 [2.97664]
taking ℎ = 0.2. Given 𝑑𝑥 = 𝑥 + 𝑦 𝑎𝑛𝑑 𝑦(0) = 1.
Using Euler’s method find approximate value of
𝑑𝑦
2. 𝑦 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑡𝑜 𝑓𝑜𝑢𝑟 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑝𝑙𝑎𝑐𝑒𝑠 𝑓𝑜𝑟 𝑥 = 0.1 Given 𝑑𝑥 = [0.9113]
𝑥−𝑦 2 𝑎𝑛𝑑 𝑦(0) = 1. Take ℎ = 0.2.
Using Euler’s method find approximate value of 𝑦 𝑎𝑡 𝑥 = 2 in five steps
3. 𝑑𝑦 𝑦−𝑥 [2.7088]
taking ℎ = 0.2. Given 𝑑𝑥 = 𝑥 𝑎𝑛𝑑 𝑦(1) = 2.
Using Euler’s method find approximate value of 𝑦 𝑎𝑡 𝑥 = 1 in five steps
4. 𝑑𝑦 [2.9186]
taking ℎ = 0.2. Given = 𝑥𝑦 𝑎𝑛𝑑 𝑦(0) = 2.
𝑑𝑥

Euler’s Modified Method / Runge-Kutta Method of Second Degree

Use Euler’s modified method to find the value of y satisfying the equation
𝑑𝑦 (𝑖) 𝑦(0.05) = 1.1635
5. = log(𝑥 + 𝑦) , 𝑦(1) = 2 for (𝑖) 𝑥 = 1.2 and (𝑖𝑖) 𝑥 = 1.4 correct to four [
(𝑖𝑖) 𝑦(0.1) = 1.3548
]
𝑑𝑥
decimal places by taking ℎ = 0.05.
Use Euler’s modified method to find the value of y satisfying the equation
𝑑𝑦 (𝑖) 𝑦(1.2) = 2.2332
6. = 𝑥 + 3𝑦, 𝑦(0) = 1 for (𝑖) 𝑥 = 0.05 and (𝑖𝑖) 𝑥 = 0.1 correct to four [
(𝑖𝑖)𝑦(1.4) = 2.4924
]
𝑑𝑥
decimal places by taking ℎ = 0.2.
Use Euler’s modified method to find the value of y satisfying the equation
𝑑𝑦 (𝑖) 𝑦(1.4) = 2.3625
7. = 2 + √𝑥𝑦, 𝑦(1.2) = 1.6403 for (𝑖) 𝑥 = 1.4 and (𝑖𝑖) 𝑥 = 1.6 correct [
(𝑖𝑖)𝑦(1.6) = 3.1695
]
𝑑𝑥
to four decimal places by taking ℎ = 0.2.
Runge-Kutta Method/ Runge-Kutta Method of Fourth order
Use Runge-Kutta method to find an approximate value of y when 𝑥 = 0.2
8. 𝑑𝑦 [𝑦(0.2) = 1.2428]
given that 𝑑𝑥 = 𝑥 + 𝑦 𝑤ℎ𝑒𝑛 𝑦 = 1 𝑎𝑡 𝑥 = 0.
Use Runge-Kutta method to find an approximate value of y when 𝑥 = 0.2
9. 𝑑𝑦 [𝑦(0.2) = 2.4432]
given that 𝑑𝑥 = 𝑥 3 + 𝑦 𝑤ℎ𝑒𝑛 𝑦 = 2 𝑎𝑡 𝑥 = 0.

12 | P a g e First Year Engineering | Applied Mathematics-II | Handbook

 SIES Graduate School of Technology

Use Runge-Kutta method to find an approximate value of y when 𝑥 = 1.2 (𝑖) 𝑦(1.1) = 1.1272
10. 𝑑𝑦 [ ]
given that 𝑑𝑥 = 𝑥𝑦 𝑤ℎ𝑒𝑛 𝑦 = 1 𝑎𝑡 𝑥 = 1 taking ℎ = 0.1 (𝑖𝑖)𝑦(1.2) = 1.2374

Use Runge-Kutta method to find an approximate value of y when 𝑥 = 1 (𝑖) 𝑦(0.5) = 1.3571
11. 𝑑𝑦 1 [ ]
given that 𝑑𝑥 = (𝑥+𝑦) 𝑤ℎ𝑒𝑛 𝑦 = 1 𝑎𝑡 𝑥 = 0 taking ℎ = 0.5 (𝑖𝑖)𝑦(1.0) = 1.5837

Numerical Integration
1 𝑥2
Find the value of the integral ∫0 𝑑𝑥, using (i) Trapezoidal rule and (ii) [
(𝑖) 0.2311
]
12. 1+𝑥 3
(𝑖𝑖)0.2311
Simpson’s (1/3)rd rule.
𝜋

13. Evaluate ∫02 √𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 𝑑𝑥 by Simpson’s (3/8)th rule by dividing into [1.7702]
six intervals.
4
Find the approximate value of ∫0 𝑒 𝑥 𝑑𝑥 by using (i) Trapezoidal and (ii) [
(𝑖) 57.9905
]
14. (𝑖𝑖)53.5961
Simpson’s (1/3)rd rule.
1 1 (𝑖) 1.5615
Find the value of the integral ∫−1 1+𝑥 2 𝑑𝑥, using (i) Trapezoidal rule, (ii)
15. [ (𝑖𝑖)1.5709 ]
Simpson’s (1/3)rd rule, (iii) Simpson’s (3/8)th rule (𝑖𝑖𝑖)1.5692
1.4 (𝑖) 4.0715
Compute the values of the integral ∫0.2 (𝑠𝑖𝑛𝑥 − 𝑙𝑜𝑔𝑒 𝑥 + 𝑒 𝑥 ) 𝑑𝑥 , using (i) [ (𝑖𝑖)4.0521 ]
16. Trapezoidal rule, (ii) Simpson’s (1/3)rd rule, (iii) Simpson’s (3/8)th rule (𝑖𝑖𝑖)4.0530

13 | P a g e First Year Engineering | Applied Mathematics-II | Handbook