QUESTION BANK FOR Calculus & Differential equations
Course CODE – 21MAT11
Course Outcomes
On completion of this course, students are able to:
CO1: Apply the knowledge of calculus to solve problems related to polar curves and its applications in
determining the bentness of a curve.
CO2: Lean the notion of partial differentiation to calculate rates of change of multivariate functions and
solve problems related to composite functions and Jacobians.
CO3: Solve first order linear/nonlinear differential equation analytically using standard methods.
CO4: Demonstrate various models through higher order differential equations and solve such linear
ordinary differential equations.
CO5: Make use of matrix theory for solving system of linear equations and compute eigen values and
eigenvectors required for matrix diagonalization process.
Prepared By
Dr. Lakshminarayanachari. K
Dr. Bhaskar. C
Prof. Naveena gn
Dr. Arunkumar. R
Dr. Madhura. K
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
MODULE – I: DIFFERENTIAL CALCULUS – I
1) With usual notation, prove that tan   r
d
dr
[Aug 2021, June 2019, 2018, Dec 2018, 2019, 2017]
1
1
1
𝑑𝑟 2
2) With usual notations prove that, the pedal equation of the form 𝑝2 = 𝑟 2 + 𝑟 4 (𝑑𝜃)
[Feb 2021, Aug 2020, Dec 2017]
3) Find the angle between the radius vector and the tangent for the following curves:
i) 𝑟 = 𝑎(1 − cos 𝜃) at 𝜃 = 𝜋⁄3
iii) 𝑟 𝑚 = 𝑎𝑚 (cos 𝑚𝜃 + sin 𝑚𝜃)
[Dec 2017]
ii) 𝑟𝑠𝑒𝑐 2 (𝜃⁄2) = 2𝑎
iv) 𝑟 2 = 𝑎2 sin 2𝜃
4) Find the angle of intersection of the following curves:
𝑎𝜃
𝑎
i) 𝑟 = 𝑎(1 + sin 𝜃) & 𝑟 = 𝑎(1 − 𝑠𝑖𝑛𝜃) [June 2017]
ii) 𝑟 = 1+𝜃 & 𝑟 = 1+𝜃2
[June 2016]
2
2
2
2
iii) 𝑟 = 2𝑠𝑖𝑛𝜃 & 𝑟 = 2(𝑠𝑖𝑛𝜃 + 𝑐𝑜𝑠𝜃) [June 2017]
iv) 𝑟 sin 2𝜃 = 𝑎 & 𝑟 cos 2𝜃 = 𝑏
v) 𝑟(1 + 𝑐𝑜𝑠𝜃) = 𝑎 & 𝑟(1 − 𝑐𝑜𝑠𝜃) = 𝑏 [Dec 2016]
vi) 𝑟 𝑛 = 𝑎𝑛 (sin 𝑛𝜃 + cos 𝑛𝜃) & 𝑟 𝑛 = 𝑎𝑛 sin 𝑛𝜃
vii) 𝑟 𝑛 = 𝑎𝑛 cos 𝑛𝜃 & 𝑟 𝑛 = 𝑏 𝑛 sin 𝑛𝜃 [June 2018, Dec 2018]
viii) 𝑟 = 𝑎𝑙𝑜𝑔𝜃 & 𝑟 = 𝑎/𝑙𝑜𝑔𝜃
[Aug 2021, Dec 2017, 2016, 2014, 2012]
ix) 𝑟 2 sin 2𝜃 = 4 & 𝑟 2 = 16𝑠𝑖𝑛2𝜃
[Dec 2015, 2012, June 2015]
2
2
x) 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃) & 𝑟 = 𝑎 cos 2𝜃
[Dec 2016, June 2016]
xi) 𝑟 = 𝑎(1 + sin 𝜃) & 𝑟 = 𝑎(1 − cos 𝜃)
[Dec 2017, June 2016]
xii) 𝑟 = 𝑎(1 + cos 𝜃) & 𝑟 = 𝑏(1 − cos 𝜃) [June 2019, 2018, 2016, 2015, 2009, Dec 2019, Dec 2017, 2015]
xiii) 𝑟 = sin 𝜃 + cos 𝜃 & 𝑟 = 2 sin 𝜃
[Dec 2019]
xiv) 𝑟 = 6 cos 𝜃 & 𝑟 = 2(1 + cos 𝜃)
[Aug 2020]
xv) 𝑟 = 2 sin 𝜃 & 𝑟 = 2 cos 𝜃
[Feb 2021]
5) Find the pedal equation for the following curves:
i) 𝑟 𝑚 cos 𝑚𝜃 = 𝑎𝑚
[June 2016] ii) 𝑟 = 2𝑎⁄1 + cos 𝜃
[June 2018, 2011, Dec 2016]
𝑛
iii) 𝑟 = sec ℎ 𝑛𝜃
[Dec 2014] iv) 𝑟 = 𝑎 + 𝑏 cos 𝜃
[June 2017]
𝑙
3
v) 𝑟 = 𝑎 sin 3𝜃
vi) 𝑟 = 1 + 𝑒 cos 𝜃
vii) 𝑟(1 − 𝑐𝑜𝑠𝜃) = 2𝑎
[Aug 2020, Dec 2012] viii) 𝑟 𝑛 = 𝑎(1 + 𝑐𝑜𝑠𝑛𝜃)
[Dec 2015]
2𝑎
2
2
ix) 𝑟 = 1 − sin 𝜃
x) 𝑟 = 𝑎 𝑐𝑜𝑠2𝜃
[Aug 2021, June 2014]
𝑛
𝑛
𝑛
𝑛
xi) 𝑟 sec 𝑛𝜃 = 𝑎 or 𝑟 = 𝑎 cos 𝑛𝜃
[June 2019, Dec 2019, June 2017, Dec 2016, 2015, 2013]
𝜃𝑐𝑜𝑡𝛼
xii) 𝑟 = 𝑎𝑒
[Dec 2018]
xiii) 𝑟 = 𝑎(1 + cos 𝜃)
[Aug 2021, June 2018, 2016, Dec 2017]
𝑚
𝑚 (cos
xiv) 𝑟 = 𝑎
𝑚𝜃 + sin 𝑚𝜃)
[Feb 2021, Dec 2016, 2011, June 2010]
6) Find the pedal equation of 𝑟 𝑛 = 𝑎𝑛 sin 𝑛𝜃 + 𝑏 𝑛 cos 𝑛𝜃 in the form 𝑝2 (𝑎2𝑛 + 𝑏 2𝑛 ) = 𝑟 2𝑛+2 .
[June 2015]
7) Derive Radius of Curvature in the Cartesian form.
2
8) Show that the radius of curvature of the curve 𝑦 = 𝑐 𝐶𝑜𝑠ℎ(𝑥⁄𝑐) is y
, where c is a constant.
c
9) Find the radius of curvature of the parabola 𝑦 2 = 4𝑎𝑥 at the point (a, a).
[Aug 2020, Dec 2016]
𝑎 𝑎
10) Find the radius of curvature of the curve √𝑥 + √𝑦 = √𝑎 at (2 , 2) .
[Dec 2017]
𝜋
11) Find the radius of curvature of the curve 𝑦 = 4𝑠𝑖𝑛𝑥 − 𝑠𝑖𝑛2𝑥 at 𝑥 = ⁄2 .
[Dec 2017]
𝑥
𝑥
12) Show that the radius of curvature of the curve 𝑦 = 𝑎𝑙𝑜𝑔𝑠𝑒𝑐 (𝑎) is 𝑎𝑠𝑒𝑐 (𝑎)
[Dec 2018, 2019]
13) Find the radius of curvature for the curve 𝑥 4 + 𝑦 4 = 2 𝑎𝑡 (1,1).
[Dec 2016, June 2016]
3𝑎 3𝑎
3
3
14) Find the radius of curvature for the Folium De-Cartes 𝑥 + 𝑦 = 3𝑎𝑥𝑦 at the point ( 2 , 2 ) on it.
[Aug 2021, June 2017, 2016, Dec 2015, 2014]
1
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
3 3
−3
15) Show that the radius of curvature of the curve 𝑥 3 + 𝑦 3 = 3𝑥𝑦 at the point (2 , 2) is 8√2 .
[Dec 2014]
2
2
2)
16) Find the radius of curvature for the curve 𝑥 𝑦 = 𝑎(𝑥 + 𝑦 at the point (−2𝑎, 2𝑎).
[Dec 2017, 2012]
2
3
3
17) Find the radius of curvature for the curve 𝑎 𝑦 = 𝑥 − 𝑎 at the point where the curve cuts 𝑥-axis.
[June 2019, Dec 2010]
18) Find the radius of curvature for the curve 𝑦 2 =
4𝑎2 (2𝑎−𝑥)
𝑥
𝑎2 (𝑎−𝑥)
19) Find the radius of curvature of the curve 𝑦 2 =
𝑥
3
where the curve meets the 𝑥-axis.
at the point (𝑎, 0).
[June 2018, Dec 2017, 2013]
20) Find the radius of curvature of the curve 𝑥𝑦 2 = 𝑎 − 𝑥 3 at (𝑎, 0).
2
2𝜌 ⁄3
𝑎𝑥
21) If 𝑦 = 𝑎+𝑥 , then show that ( 𝑎 )
𝑦 2
[June 2018]
𝑥 2
= (𝑥 ) + (𝑦) where 𝜌 is the radius of curvature at (𝑥, 𝑦)
[Feb 2021, June 2017, Dec 2016]
𝑥
𝑦
22) Prove that the radius of curvature at any point (𝑥, 𝑦) on the curve √𝑎 + √𝑏 =1 is 𝜌 =
23) Find the radius of curvature for the curve 𝑥 = 𝑎𝑙𝑜𝑔(sec 𝑡 + tan 𝑡) , 𝑦 = 𝑎 sec 𝑡
𝜋
2(𝑎𝑥+𝑏𝑦)3/2
𝑎𝑏
. [June 2011]
[July 2015]
24) Find the radius of curvature for the Astroid 𝑥 = 𝑎𝑐𝑜𝑠 𝜃, 𝑦 = 𝑎𝑠𝑖𝑛 𝜃 𝑎𝑡 𝜃 = 4
25) Find the radius of curvature of the curve 𝑥 = (cos 𝑡 + 𝑡 sin 𝑡), 𝑦 = 𝑎(sin 𝑡 - 𝑡 cos 𝑡)
𝑡
26) Find the radius of curvature of the curve 𝑥 = 𝑎 [cos 𝑡 + log tan 2] 𝑦 = 𝑎 sin 𝑡
[Aug 2020, July 2015]
27) Show that the radius of curvature at any point 𝜃 on the Cycloid 𝑥 = 𝑎(𝜃 + sin 𝜃) , 𝑦 = 𝑎(1 − cos 𝜃) is
𝜃
4𝑎 cos 2
[July 2012]
28) Derive Radius of Curvature in the Polar form.
29) Find the radius of curvature to the curve 𝑟 = 𝑎 sin 𝑛𝜃 at the pole.
[June 2018]
𝑛
𝑛
𝑛−1
30) Show that the radius of curvature of the curve 𝑟 = 𝑎 cos 𝑛𝜃 varies inversely as 𝑟 .
[Dec 2016]
31) Find the radius of curvature of the curve 𝑟 = 𝑎(1 + cos 𝜃).
[Dec 2018, 2017]
32) Show that for the curve 𝑟(1 − cos 𝜃) = 2𝑎, 𝜌2 varies as 𝑟 3 .
[June 2019]
𝑛
𝑛
33) Find the radius of curvature of the curve 𝑟 = 𝑎 sin 𝑛𝜃.
[Dec 2017, 2015]
34) Find the radius of curvature of the polar curve 𝑟 2 = 𝑎2 cos 2𝜃.
[June 2017]
35) Find the radius of curvature of the curve 𝑟 = 𝑎(1 − cos 𝜃).
3
36) Find the radius of curvature for the Parabola
2𝑎
𝑟
3
= 1 + cos 𝜃
[Feb 2021]
𝜃
37) Show that for the curve 𝑟𝑐𝑜𝑠 2 ( 2) = 𝑎, 𝜌2 varies as 𝑟 3 .
38) Show that the radius of curvature of the curve 𝑝𝑎2 = 𝑟 3 is
2
𝑟3
𝜌2
39) Show that for the curve 𝑝 = 2𝑎 , 𝑟 is a constant.
40) Show that for the curve 𝑝2 = 𝑎𝑟, 𝜌2 varies as 𝑟 3 .
41) Show that for the ellipse in pedal form,
1
𝑝2
1
1
𝑎2
3𝑟
𝑟2
= 𝑎2 + 𝑏2 − 𝑎2 𝑏2 , the radius of curvature at the point (p, r) is
𝑎2 𝑏2
𝑝3
.
Self-Study:
42) Find the coordinates of the center of curvature at any point of the parabola 𝑥 2 = 4𝑎𝑦. Hence find its evolute.
43) Show that the evolute of the parabola 𝑦 2 = 4𝑎𝑥 is 27𝑎𝑦 2 = 4(𝑥 − 2𝑎)3 .
[Aug 2021, 2020, Feb 2021, Dec 2019, June 2019, Dec 2018]
44) Show that the evolute of the cycloid 𝑥 = 𝑎(𝜃 − 𝑠𝑖𝑛𝜃), 𝑦 = 𝑎(1 − 𝑐𝑜𝑠𝜃) is another equal cycloid.
45) Find the evolute of the curve 𝑥 = 𝑎𝑐𝑜𝑠 3 𝜃, 𝑦 = 𝑎𝑠𝑖𝑛3 𝜃, i,e 𝑥 2/3 + 𝑦 2/3 = 𝑎2/3 .
𝑡
𝑥
46) Show that the evolute of the curve 𝑥 = 𝑎 [cos 𝑡 + 𝑙𝑜𝑔 tan (2)] , 𝑦 = 𝑎 sin 𝑡 is 𝑦 = 𝑎 cosh (𝑎).
[Aug 2020]
2
47) Find the evolute of the curve 𝑥𝑦 = 𝑐 .
𝑥2
𝑦2
48) Find the evolute of the ellipse 𝑎2 + 𝑏2 = 1.
**************************************************
2
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
MODULE – II : DIFFERENTIAL CALCULUS – II
Taylor’s and Maclaurin’s series:
1.
2.
3.
4.
5.
6.
Obtain the Taylors series expansion of 𝑙𝑜𝑔(𝑐𝑜𝑠𝑥) about the point at 𝑥 = 𝜋/3 up to the 4th degree term
Expand 𝑡𝑎𝑛𝑥 about the point 𝑥 = 𝜋/4 upto the third degree term.
Expand tan−1 𝑥 in powers of (𝑥 − 1) up to the term containing fourth degree.
Find Maclaurin series expansion of 𝑠𝑒𝑐𝑥 upto 4th degree term.
[Dec 2015]
5
Using Maclaurin’s series expand 𝑦 = 𝑡𝑎𝑛𝑥 up to the term containing 𝑥 [Aug 2020, Dec 2019, June 2016, 2015]
Expand tan−1 (1 + 𝑥) in Maclaurin’s expansion upto first three non –vanishing terms.
7. Expand 𝑒 tan
−1 𝑥
in Maclaurin’s expansion upto first three non –vanishing terms.
x 2 x3 x 4
 

2
6 24
[July 2021, Aug 2020, June 2018, Dec 2018, 2015, 2014]
8. Using Maclaurin’s theorem prove that 1  sin 2 x  1  x 
9. Using Maclaurin’s theorem prove that √1 + cos 2𝑥 = √2 [1 −
𝑥2
2
𝑥4
+ 24 − ⋯ ]
[Jan 2021]
10. Expand 𝑙𝑜𝑔(1 + 𝑐𝑜𝑠𝑥) by the Maclaurin’s series up to the term containing 𝑥 4 .
[June 2019, 2017, Dec 2017]
4
11. Expand 𝑙𝑜𝑔(1 + 𝑠𝑖𝑛𝑥) by the Maclaurin’s series up to the term containing 𝑥 .
[Dec 2012]
6
12. Expand 𝑙𝑜𝑔(𝑠𝑒𝑐𝑥)by the Maclaurin’s series up to the term containing 𝑥
[Dec 2019, 2017, 2010, June 2017, 2016, 2012]
𝑥
13. Expand 𝑙𝑜𝑔(1 + 𝑒 ) using Maclaurin’s series up to and including 3rd degree term. [June 2016, Dec 2016, 2013]
14. Expand 𝑙𝑜𝑔(1 + 𝑥) using Maclaurin’s series up to the term containing 𝑥 4 .
[Dec 2017]
𝑠𝑖𝑛𝑥
4
15. Expand 𝑒
by the Maclaurin’s series up to the term containing 𝑥 .
[Dec 2017]
𝑥
4
16. Obtain Maclaurin series for 𝑒 𝑐𝑜𝑠𝑥 upto the term containing 𝑥
[June 2018]
𝑒𝑥
17. Expand 1+𝑒 𝑥 using Maclaurin’s series up to and including 3rd degree term.
[Dec 2016]
18. Find the first four non zero terms in the expansion of 𝑓(𝑥) = 𝑒 𝑥 −1
[Dec 2014]
𝑥
Evaluate the following Indeterminate forms:
19. lim𝑥→𝜋 (𝑠𝑖𝑛𝑥)
𝑡𝑎𝑛𝑥
20.
2
𝑥
21. lim𝑥→0 (𝑎 + 𝑥)
1
𝑥
[Dec 2016]
1
𝑎𝑥 +𝑏 𝑥 𝑥
23. lim𝑥→0 (
)
2
1
[Aug 2020, Dec 2017]
𝑡𝑎𝑛𝑥 2
lim𝑥→0 ( 𝑥 )𝑥
1
2𝑥 +3𝑥 +4𝑥 𝑥
22. lim𝑥→0 (
)
3
24. lim𝑥→0 (𝑐𝑜𝑠𝑥)𝑐𝑜𝑡
1
25.
[Dec 2016, June 2016]
2𝑥
1 2 sin 𝑥
𝑎+𝑥 𝑥
lim𝑥→0 (𝑎−𝑥)
26. lim𝑥→0 (𝑥)
[Dec 2010]
[June 2017]
[Jan 2021]
𝜋𝑥
27. lim (2 −
𝑥→𝑎
𝑥 𝑡𝑎𝑛( 2𝑎)
)
𝑎
[Dec 2017, June 2016] 28. lim𝑥→𝜋 (𝑡𝑎𝑛𝑥)𝑐𝑜𝑠𝑥
2
1
29.
𝑠𝑖𝑛𝑥 𝑥
lim𝑥→0 ( 𝑥 )
30.
[July 2021]
32. lim𝑥→0 (
1
31. lim𝑥→0 (𝑐𝑜𝑠𝑥)𝑥2
33. lim𝑥→0 (
1
𝑎𝑥 +𝑏 𝑥 +𝑐 𝑥 𝑥
)
3
𝑥
)
[June 2017]
[Aug 2020]
[Jan 2021, Aug 2020, Dec 2019, June 2019]
1
𝑎𝑥 +𝑏 𝑥 +𝑐 𝑥 +𝑑𝑥 𝑥
34. lim𝑥→0 (
1
𝑠𝑖𝑛𝑥 2
lim𝑥→0 ( 𝑥 )𝑥
1
tan 𝑥 ⁄𝑥
[Dec 2014]
[Dec 2012]
4
)
[July 2021, June 2018, 2017, Dec 2018, 2015, 2012]
3
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
Partial Differentiation:
35. If 𝑢 = 𝑥 𝑦 , then verify that
 2u  2u

.
xy yx


1 / 2
 2u  2u  2u
36. If u  x 2  y 2  z 2
, then prove that


 0.
x 2
y 2
z 2
37. If 𝑢 = 𝑥 2 𝑦 + 𝑦 2 𝑧 + 𝑧 2 𝑥, then show that 𝑢𝑥 + 𝑢𝑦 + 𝑢𝑧 = (𝑥 + 𝑦 + 𝑧)2
𝜕𝑧
𝜕𝑧
38. If 𝑧 = 𝑒 𝑎𝑥+𝑏𝑦 𝑓(𝑎𝑥 − 𝑏𝑦) , show that 𝑏 𝜕𝑥 + 𝑎 𝜕𝑦 = 2𝑎𝑏𝑧
Total derivative and Chain rule:
39. If 𝑢 = 𝑥 3 𝑦 2 + 𝑥 2 𝑦 3 where 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡, find
𝑑𝑢
𝑑𝑡
.
[Dec 2016]
𝑥
𝑑𝑢
𝑦
𝑑𝑡
𝑑𝑢
40. If 𝑢 = 𝑡𝑎𝑛−1 (𝑦) where 𝑥 = 𝑒 𝑡 − 𝑒 −𝑡 , 𝑦 = 𝑒 𝑡 + 𝑒 −𝑡 , find
41. If 𝑢 = 𝑡𝑎𝑛−1 (𝑥 ) where 𝑥 = 𝑒 𝑡 − 𝑒 −𝑡 , 𝑦 = 𝑒 𝑡 + 𝑒 −𝑡 , find
.
[June 2017]
.
[June 2018]
𝑑𝑡
2𝑡
42. If 𝑢 = 𝑥 2 + 𝑦 2 + 𝑧 2 where 𝑥 = 𝑒 2𝑡 , 𝑦 = 𝑒 2𝑡 𝑐𝑜𝑠2𝑡, 𝑧 = 𝑒 𝑠𝑖𝑛2𝑡, find
43. If 𝑢 = log(𝑥 + 𝑦 + 𝑧) where 𝑥 = 𝑒 −𝑡 , 𝑦 = 𝑠𝑖𝑛𝑡, 𝑧 = 𝑐𝑜𝑠𝑡, find
𝑢
44. If 𝑧 = 𝑓(𝑥, 𝑦) where 𝑥 = 𝑒 − 𝑒
−𝑣
,𝑦 = 𝑒
−𝑢
𝑑𝑢
𝑑𝑡
.
[Dec 2015]
𝑑𝑡
𝜕𝑧
𝑣
𝑑𝑢
𝜕𝑧
𝜕𝑧
𝜕𝑧
− 𝑒 prove that 𝜕𝑢 − 𝜕𝑣 = 𝑥 𝜕𝑥 − 𝑦 𝜕𝑦
2
1  z 
 z   z 
 z 
45. If 𝑧 = 𝑓(𝑥, 𝑦) where 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃 prove that          2  
r   
 x   y 
 r 
2
2
[Dec 2015]
2
[June 2018, 2017, Dec 2017, 2014]
yx zx
u
u
u
 , find the value of x 2
,
 y2
 z2
zx 
x
y
z
 xy
46. If u  u
47. If 𝑢 = 𝑓 (𝑦 − 𝑧, 𝑧 − 𝑥, 𝑥 − 𝑦) then prove that
𝜕𝑢
𝜕𝑢
𝜕𝑥
𝜕𝑢
𝜕𝑢
[Dec 2011]
𝜕𝑢
+ 𝜕𝑦 + 𝜕𝑧 = 0
𝜕𝑢
48. If 𝑢 = 𝑓(𝑥 − 𝑦, 𝑦 − 𝑧, 𝑧 − 𝑥), show that 𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧 = 0
𝑥 𝑦 𝑧
𝜕𝑢
𝜕𝑢
[Jan 2021, Dec 2018, 2017, June 2015]
[July 2021, Aug 2020, Dec 2019]
𝜕𝑢
49. If 𝑢 = 𝑓 (𝑦 , 𝑧 , 𝑥) then prove that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 + 𝑧 𝜕𝑧 = 0 [June 2019, 2018, 2017, 2016, 2014, Dec 2019, 2016]
50. If 𝑢 = 𝑓 (2𝑥 − 3𝑦, 3𝑦 − 4𝑧, 4𝑧 − 2𝑥) then find
𝑦
𝜕𝑢
𝜕𝑢
1 𝜕𝑢
2 𝜕𝑥
1 𝜕𝑢
1 𝜕𝑢
+ 3 𝜕𝑦 + 4 𝜕𝑧
[June 2017, Dec 2013]
𝜕𝑢
51. If 𝑢 = 𝑓 (𝑥𝑧, 𝑧 ) prove that 𝑥 𝜕𝑥 − 𝑦 𝜕𝑦 − 𝑧 𝜕𝑧 = 0
Maxima and Minima:
52. Find the extreme values of f ( x, y)  x3  3xy 2  15x 2  15 y 2  72 x
53. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 − 3𝑥 − 12𝑦 + 20
[Aug 2020, June 2017]
[June 2019, Dec 2019, June 2018]
54. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑥 3 𝑦 2 (1 − 𝑥 − 𝑦)
[Dec 2012]
55. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦, 𝑎 > 0
[June 2011]
56. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑥 4 + 𝑦 4 − 2(𝑥 − 𝑦)2
[June 2015]
57. Examine the function 𝑥𝑦(𝑎 − 𝑥 − 𝑦) for extreme values.
[June 2016]
4
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
58. Examine the function 𝑓(𝑥, 𝑦) = 1 + sin(𝑥 2 + 𝑦 2 ) for extreme values.
[Dec 2012]
59. Find the extreme values of 𝑓(𝑥, 𝑦) = 𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛𝑦 + sin(𝑥 + 𝑦)
[June 2016]
60. Examine the function 𝑓(𝑥, 𝑦) = 2 + 2𝑥 + 2𝑦 − 𝑥 2 − 𝑦 2 for its extreme values.
[Jan 2021]
𝒂 𝒂
61. Show that the function 𝑥𝑦(𝑎 − 𝑥 − 𝑦) is maximum at (𝟑 , 𝟑). Hence find maximum value if 𝑎 > 0.
[July 2021]
Jacobians:
 ( x, y )
 (u, v)
 (r , )
63. If 𝑥 = 𝑟𝑐𝑜𝑠𝜃 and 𝑦 = 𝑟𝑠𝑖𝑛𝜃 find
 ( x, y )
62. If 𝑥 = 𝑢 (1 − 𝑣), 𝑦 = 𝑢 𝑣 then evaluate
[Dec 2011]
64. If 𝑢 = 𝑥 2 − 𝑦 2 , 𝑣 = 2𝑥𝑦 and 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃 determine the value of
65. If 𝑥 = 𝑟𝑠𝑖𝑛𝜃𝑐𝑜𝑠∅, 𝑦 = 𝑟𝑠𝑖𝑛𝜃𝑠𝑖𝑛∅ and 𝑧 = 𝑟𝑐𝑜𝑠𝜃 find
 (u, v)
 (r , )
 ( x, y , z )
 ( r , ,  )
[June 2018, 2016, Dec 2013]
 (u, v, w)
where 𝑢 = 𝑥 2 + 𝑦 2 + 𝑧 2 , 𝑣 = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥, 𝑤 = 𝑥 + 𝑦 + 𝑧
 ( x, y , z )
[July 2021, Dec 2019, Dec 2018, 2017, 2016, 2014, June 2017]
 (u, v, w) 
zx
xy
yz
  4
67. If u 
and v 
and w 
then show that J 
[Dec 2016, 2010, June 2016, 2015]
y
z
x
 ( x, y , z ) 
66. Find
68. If 𝑢 = 𝑥 + 3𝑦 + 𝑧 3 , 𝑣 = 𝑥 2 𝑦𝑧, 𝑤 = 2𝑥 2 – 𝑥𝑦, Evaluate
 (u, v, w)
at [1, −1, 0]
 ( x, y , z )
[June 2018]
 (u, v, w)
at [1, −1, 0]
 ( x, y , z )
[June 2019, Dec2018, Dec 2016, 2015]
 (u, v, w)
70. If 𝑢 = 𝑥 + 3𝑦 2 , 𝑣 = 4𝑥 2 𝑦𝑧, 𝑤 = 2𝑧 2 – 𝑥𝑦 Evaluate
at [1, −1, 0]
[Dec 2017]
 ( x, y , z )
69. If 𝑢 = 𝑥 + 3𝑦 2 − 𝑧 3 , 𝑣 = 4𝑥 2 𝑦𝑧, 𝑤 = 2𝑧 2 – 𝑥𝑦, Evaluate
71. If 𝑥 + 𝑦 + 𝑧 = 𝑢, 𝑦 + 𝑧 = 𝑢𝑣, 𝑧 = 𝑢𝑣𝑤, then evaluate
72. If 𝑥 + 𝑦 + 𝑧 = 𝑢, 𝑦 + 𝑧 = 𝑣, 𝑧 = 𝑢𝑣𝑤, find
 ( x, y, z)
 (u , v, w )
 ( x, y , z )
 (u, v, w)
[Dec 2017, 2011]
73. If 𝑢 = 3𝑥 + 2𝑦 − 𝑧 ; 𝑣 = 𝑥 − 2𝑦 + 𝑧 ; 𝑤 = 𝑥 2 + 2𝑥𝑦 − 𝑥𝑧 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
𝑥
𝑦
𝑧
74. If 𝑢 = 𝑦−𝑧 , 𝑣 = 𝑧−𝑥 , 𝑤 = 𝑥−𝑦 , find
[June 2018, 2015, Dec 2015]
𝜕(𝑢,𝑣,𝑤)
(𝑥,𝑦,𝑧)
=0
 (u, v, w)
 ( x, y , z )
[June 2017]
75. Given 𝑥 = 𝑎(𝑢 + 𝑣), 𝑦 = 𝑏(1 − 𝑣) and 𝑢 = 𝑟 2 𝑐𝑜𝑠2𝜃, 𝑣 = 𝑟 2 𝑠𝑖𝑛2𝜃 find
76. If 𝑢 = 𝑥 cos 𝑦 cos 𝑧 , 𝑣 = 𝑥 cos 𝑦 sin 𝑧 , 𝑤 = 𝑥 sin 𝑦 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
[Jan 2021]
𝜕(𝑢,𝑣,𝑤)
𝜕(𝑥,𝑦,𝑧)
 ( x, y )
 (r , )
[Dec 2017]
= −𝑥 2 cos 𝑦.
[Aug 2020]
5
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
Self-Study:
Euler’s Theorem and Problems
77. If 𝑢 = tan−1 (
𝑥 3 +𝑦 3
𝑥−𝑦
𝑥 3 +𝑦 3
78. If 𝑢 = sin−1 (
𝑥−𝑦
𝑥 4 +𝑦4
79. If 𝑢 = log (
𝑥+𝑦
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
) Prove that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = sin 2𝑢 .
) Prove that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 2 tan 𝑢 .
) Prove that 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 3 .
𝑥2𝑦2
80. If 𝑢 = sin−1 ( 𝑥+𝑦 ) Prove that 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 3 tan 𝑢 .
𝑥3𝑦3
3
81. If 𝑢 = tan−1 (𝑥 3 +𝑦 3 ) Prove that 𝑥𝑢𝑥 + 𝑦𝑢𝑦 = 2 sin 2𝑢 .
Lagrange Undetermined Multipliers with single constraint:
82. If 𝒙, 𝒚, 𝒛 are angles of a triangle, show that the maximum value of 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦𝑠𝑖𝑛𝑧 is
𝟑√𝟑
𝟖
83. Find the minimum value of 𝑥 2 + 𝑦 2 + 𝑧 2 under the condition 𝑥 + 𝑦 + 𝑧 = 3𝑎.
[July 2021, June 2017]
[Dec 2018]
84. Find the maximum value of 𝒙𝒚 subject to the condition 𝑥 2 + 𝑥𝑦 + 𝑦 2 = 𝑎2 .
85. The temperature T at any point (𝑥, 𝑦, 𝑧) in space in 𝑇 = 400𝑥𝑦𝑧 2 . Find the highest temperature on the surface of
the unit sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 1.
[Jan 2021]
86. A rectangular box open at top is to have a volume of 32 cubic feet. Find the dimensions of the box, if the total
surface area is minimum.
[June 2019, Dec 2018]
********************************************
6
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
MODULE – III : ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER
Solve the following Differential Equations:
dy
 xy 3  xy
[June 2016]
2.  x 2  y 2  x  dx  xy dy  0 [July 2021, June 2018, 2017]
dx
dy
dy
 y tan x  y 3 sec x [July 2021, Jan 2021, June 2019, Dec 2017] 4.
 y tan x  y 2 sec x [Dec 2018, 2014, June 2016]
3.
dx
dx
dy y cos x  sin y  y
dy

 0 [Jan 2020, Dec 2018, 2017, 2015] 6. xy (1  xy 2 )
5.
 1 [June 2018, 2017, 2015, Dec 2017]
dx sin x  x cos y  x
dx
1.
7. ye xy dx  ( xe xy  2 y)dy  0
dy
9. x  y  x 3 y 6
dx
[Dec 2017]
8.
[Dec 2017]
dy
sin x cos 2 x
10.
 y tan x 
dx
y2
[Dec 2015]
11. (2 x log x  xy )dy  2 ydx  0
dy 2
y2
 y 3
dx x
x
[Dec 2016]
12. ( x 2  y 3  6 x)dx  y 2 xdy  0 [June 2016, 2015]
[Dec 2016]
dy y
  y2x
dx x
 dy 
15. (4 xy  3 y 2  x)dx  x( x  2 y)dy  0 [Jan 2020, Dec 2015] 16. e y  1  e x
 dx 
13. (2 x 3  xy 2  2 y  3)dx  ( x 2 y  2 x)dy  0
[Dec 2017]
[Dec 2017]
14.
[Dec 2015]
17. (1  2 xy cos x 2  2 xy )dx  (sin x 2  x 2 )dy  0 [June 2015] 18. ( xy 3  y)dx  2( x 2 y 2  x  y 4 )dy  0 [Dec 2014]
19. r sin   cos
dr
 r2
d
[June 2018]
21. ( y 2 e xy  4 x 3 )dx  (2 xye xy  3 y 2 )dy  0 [June 2017]
2
23. (1  e
x
y
2
x
x
)dx  e y (1  )dy  0
y
𝑑𝑟
25. 𝑟 sin 𝜃 − cos 𝜃 𝑑𝜃 = 𝑟 2
[June 2019]
20.
dy
 x sin 2 y  x 3 cos 2 y
dx



[June 2017]

22. y 4  2 y dx  xy 3  2 y 4  4 x dy  0
𝑑𝑦
24. 𝑥 3 𝑑𝑥 − 𝑥 2 𝑦 = −𝑦 4 cos 𝑥
[Aug 2020]
[Jan 2020]
26. [4𝑥 3 𝑦 2 + 𝑦 cos(𝑥𝑦)]𝑑𝑥 + [2𝑥 4 𝑦 + 𝑥 cos(𝑥𝑦)]𝑑𝑦 = 0
1
27. Solve [𝑦 (𝑥 + 𝑥) + 𝑐𝑜𝑠𝑦] 𝑑𝑥 + [𝑥 + 𝑙𝑜𝑔𝑥 − 𝑥𝑠𝑖𝑛𝑦]𝑑𝑦
[Aug 2020]
[Jan 2021]
28. (cos x tan y  cos( x  y))dx  (sin x sec 2 y  cos( x  y)dy  0
ORTHOGONAL TRAJECTORIES:
1.
2.
3.
4.
5.
Find the orthogonal trajectories of a family of circles 𝑟 = 2𝑎𝑐𝑜𝑠𝜃, where ‘𝑎’ is a parameter.
[June 2017]
𝑛
𝑛
Find the orthogonal trajectories of the curve 𝑟 𝑐𝑜𝑠 𝑛𝜃 = 𝑎
[June 2019, Dec 2016]
Find the orthogonal trajectories of the curve 𝑟 𝑛 = 𝑎𝑛 𝑐𝑜𝑠 𝑛𝜃
[Dec 2017, 2015 June 2016]
Find the orthogonal trajectories of the curve 𝑟 𝑛 = 𝑎𝑛 𝑠𝑖𝑛 𝑛𝜃, where ‘𝑎’ is a parameter. [June 2017, Dec 2015]
Find the orthogonal trajectories of the curve 𝑟 𝑛 𝑠𝑖𝑛 𝑛𝜃 = 𝑎𝑛 , where ‘𝑎’ is a parameter.
[June 2018]
7
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
𝜃
6. Show that orthogonal trajectories of the families of cardioides 𝑟 = 𝑎𝑐𝑜𝑠 2 ( 2) is another family of cardioides
𝜃
𝑟 = 𝑏𝑠𝑖𝑛2 ( 2)
[Dec 2014]
2a
 1  cos 
r
8. Find the orthogonal trajectories of the curve r = 4 a sec θ tan θ
[Dec 2016, June 2015]
9. Find the orthogonal trajectories of the family of cardioides 𝑟 = 𝑎(1 + 𝑐𝑜𝑠𝜃)
[Aug 2020]
2

k 
10. Find the orthogonal trajectories of the family of curves  r   cos  =a, a being a parameter.
[July 2015]
r 

7. Find the orthogonal trajectories of a family
11. Show that the family of parabolas 𝑦 2 = 4𝑎(𝑥 + 𝑎) is self-orthogonal.
[Jan 2021, Dec 2018, June 2018, 2016]
12. Find the orthogonal trajectories of the family of the curves 𝑦 = 𝑥 + 𝑐𝑒 −𝑥 , where 𝑐 is the parameter. [Dec 2017]
13. Find the orthogonal trajectories of the curve 𝑦 = 𝑐𝑥 3
[Dec 2017]
2
2
x
y
 2
 1 , λ is a parameter, is self-orthogonal.
14. Show that the family of the curves 2
a  b 
[June 2017, Dec 2015, Dec 2012]
2
2
x
y
 1 , λ is a parameter
15. Find the orthogonal trajectories of the family of the curve 2  2
a
b 
[July 2021, June 2015]
16. Find the orthogonal trajectories of the family of the curve x 2 / 3  y 2 / 3  a 2 / 3
[Dec 2016]
2
17. Find the orthogonal trajectories of the family of the curve 𝑦 = 4𝑎𝑥
NEWTON’S LAW OF COOLING:
1. Water at temperature 10° 𝐶 takes 5 minutes to warm upto 20° 𝐶 in a room temperature 40° 𝐶. Find the
temperature after 20 minutes.
[June 2018, 2017]
2. A body originally at 80℃ cools down to 60℃ in 20 minutes. The temperature of the air being 40℃. What will
be the temperature of the body after 40 minutes from the original?
[Dec 2017, 2016, 2015, June 2017, 2015]
°
°
3. A body heated to 110 𝐶 and placed in air at 10 𝐶. After one hour its temperature becomes 60° 𝐶. How much
additional time is required for it to cool to 30° 𝐶?
[Dec 2017, 2015]
4. If the temperature of the air is 30℃ and the substance cools from 100℃ to 70℃ in 15 minutes. Find when the
temperature will be 40℃.
[June 2016, Dec 2014]
5. Water at temperature 100℃ cools down to 88℃ in 10 minutes in a room temperature of 25℃. Find the
temperature of the water after 20 minutes
[June 2018]
6. Suppose an object is heated to 300° 𝐹 and allowed to cool in a room whose temperature is 80° 𝐹. After 10
minutes the temperature of the object is 250° 𝐹. What will be its temperature after 20 minutes?
[June 2016]
°
7. A body in air at 25℃ cools from 100℃ to 75 𝐶 in 1 minute. Find the temperature of the body at the end of 3
minutes.
[July 2021, Jan 2021, Aug 2020, Dec 2018]
8. If the air is maintained at 30℃ and the temperature of the body cools from 80℃ to 60℃ in 12 minutes. Find the
temperature of the body after 24 minutes.
[June 2019]
NON LINEAR DIFFERENTIAL EQUATIONS OF FIRST ORDER:
1.
2.
3.
4.
5.
Solve 𝑃2 + 2𝑃𝑦𝑐𝑜𝑡𝑥 − 𝑦 2 = 0
Solve 𝑥𝑦(𝑃2 + 1) = (𝑥 2 + 𝑦 2 )𝑃
Solve 𝑃(𝑃 + 𝑦) = 𝑥(𝑥 + 𝑦)
Solve yp 2  ( x  y) p  x  0
Solve 𝑥𝑝2 − (2𝑥 + 3𝑦)𝑝 + 6𝑦 = 0
[Jan 2020]
[Jan 2021, Dec 2018, June 2018]
[Dec 2011, June 2011]
[July 2021, June 2019]
[Dec 2015]
8
Calculus and Differential Equations / 21MAT11
6.
Solve 𝑥𝑦𝑃2 + 𝑃(3𝑥 2 − 2𝑦 2 ) − 6𝑥𝑦 = 0
7.
Solve 𝑥 2 (𝑑𝑥 ) + 𝑥𝑦 𝑑𝑥 − 6𝑦 2 = 0
𝑑𝑦 2
𝑑𝑦
8.
Solve 𝑥 2 𝑝2 + 3𝑥𝑦𝑝 + 2𝑦 2 = 0
9.
Solve
10.
11.
12.
13.
14.
Solve
Solve
Solve
Solve
Solve
𝑑𝑦
𝑑𝑥
4
𝑑𝑥
DEPARTMENT OF MATHEMATICS
𝑥
𝑦
− 𝑑𝑦 = 𝑦 − 𝑥
𝑃 − (𝑥 + 2𝑦 + 1)𝑃3 + (𝑥 + 2𝑦 + 2𝑥𝑦)𝑃2 − 2𝑥𝑦𝑃 = 0
𝑥𝑝3 − 𝑦𝑝2 + 1 = 0 by Clairaut’s equation
(𝑃𝑥 − 𝑦)(𝑃𝑦 + 𝑥) = 𝑎2 𝑃, use the substitution 𝑋 = 𝑥 2 , 𝑌 = 𝑦 2
(𝑝𝑥 − 𝑦)(𝑝𝑦 + 𝑥) = 2𝑝 by reducing into Clairaut’s form, taking 𝑋 = 𝑥 2 , 𝑌 = 𝑦 2
(𝑝𝑥 − 𝑦)(𝑝𝑦 + 𝑥) = 0 by reducing into Clairaut’s form, taking 𝑋 = 𝑥 2 , 𝑌 = 𝑦 2
[Dec 2014, 2011]
[June 2011]
[June 2019]
[Dec 2018]
Self-Study:
L-R CIRCUITS:
1. A 12 volt battery is connected to a series circuit in which the inductance is ½ Henry and resistance is 10 ohms.
Determine the current if the initial current is zero.
[Dec 2017]
2. The R-L circuit differential equation acted by a electro motive force 𝐸𝑠𝑖𝑛𝜔𝑡, satisfies the differential equation,
di
L dt + Ri = Esinωt.if there is no current in the circuit initially, obtain the value of current at any time ‘𝑡’.
[Dec 2016]
3. A series circuit with resistance R, inductance L and electromotive force E is governed by the differential
𝑑𝑖
equation 𝐿 𝑑𝑡 = 𝑅𝑖 = 𝐸, where L and R are constants and initially the current i is zero. Find the current at any
time t.
[Jan 2021]
𝑑𝑖
4. Solve the equation 𝐿 + 𝑅𝑖 = 𝐸0 𝑠𝑖𝑛𝑤𝑡 Where L, R and 𝐸0 are constants and discuss the case when t
𝑑𝑡
increases indefinitely.
[Aug 2020, Jan 2020]
𝑑𝐼
5. Solve the L-R circuit 𝐿 𝐷𝑇 + 𝑅𝐼 = 𝐸 Initially I = 0 when t = 0
[July 2021]
6. When a switch is closed in a circuit containing a battery E, a resistance R and an inductance L. The current i
di
builds up at rate given by L dt + Ri = Esinωt. Find i as a function of t. How long will it be, before the current
has reached one half its final value if E = 6 volts, R = 100 ohms and L = 0.1 Henry.
Solvable for x and y :
1.
2.
3.
4.
Solve 𝑦 2 log 𝑦 = 𝑥𝑝𝑦 + 𝑝2
Solve 𝑝3 − 4𝑥𝑦𝑝 + 8𝑦 2 = 0
Solve 𝑥𝑝4 − 2𝑦𝑝3 + 12𝑥 3 = 0
Solve 𝑦 = 2𝑝𝑥 + 𝑝2 𝑦
**************************************
9
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
MODULE – IV: ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER
Solve the following linear differential equations:
1.
3.
5.
6.
7.
8.
d3y
d2y
dy

3
3  y  0
3
2
dx
dx
dx
d3y d2y
dy
 2  4  4y  0
3
dx
dx
dx
4. (𝐷4 − 𝑚4 )𝑦 = 0
[June 2018]
2.
(𝐷4 − 1)𝑦 = 0
[June 2010]
[June 2017]
[June 2017]
(𝐷 + 5𝐷 + 6𝐷 − 4𝐷 − 8)𝑦 = 0
[Dec 2016]
4
3
2
(4𝐷 − 4𝐷 − 23𝐷 + 12𝐷 + 36)𝑦 = 0
[Dec 2016, 2011, June 2015]
4
3
2
(4𝐷 − 8𝐷 − 7𝐷 + 11𝐷 + 6)𝑦 = 0 [Jan 2021, Jan 2020, Dec 2019, 2016, June 2018, 2016, 2015]
(𝐷4 + 4𝐷3 – 5𝐷2 – 36𝐷– 36) 𝑦 (𝑥) = 0
[Dec 2011]
4
3
2
9. ( D 4  2D 3  2D 2  2D  1) y  0
2
dx
(0)  15
10. d 2x  5 dx  6 x  0 given x(0)  0,
dt
dt
dt
2
dy
13
  for x  0.
11. 9 d 2y  6 dy  y  0 given y  4 and
dx
3
dx
dx
[Dec 2017]
2
dy d y
d2y
dy

for x  0.
 4  5 y  0 given y  2 and
2
dx dx 2
dx
dx
d2y
dy
13.
[June 2018]
14.(𝐷2 + 4𝐷 + 3)𝑦 = 𝑒 −𝑥 [Jan 2020, Dec 2019]
 6  9 y  6e 3 x
2
dx
dx
12.
d3y d2y
dy
 2  4  4 y  3e x
3
dx
dx
dx
2
d y dy
17.

 y  (1  e x ) 2
dx 2 dx
15.
19.
16.
[June 2018]
18. (𝐷2 + 4)𝑦 = 𝑥 2 + 𝑐𝑜𝑠2𝑥 + 2−𝑥 [Dec 2017]
d3y d2y
dy
 2  4  4 y  sinh(2 x  3) [Dec 2017]
3
dx
dx
dx
21. D 2  3D  2y  sin 2 x
[Dec 2017]
23. ( D  4D  4) y  e  sin x
2
2x
[June 2019]
25. (𝐷2 – 4𝐷 + 4)𝑦 = 𝑒 2𝑥 + 𝑐𝑜𝑠2𝑥 + 4 [Dec 2014]
27.
d2y
 4 y  2x
dx 2
[June 2018]
d3y
d 2 y dy
2 2 
 e  x  sin 2 x
3
dx
dx
dx
[Dec 2016, 2011]
[June 2015]
 x
20. y   2 y   y  cosh 
2
22. (𝐷3 − 1)𝑦 = 3𝑐𝑜𝑠2𝑥 [Jan 2020, Dec 2019, 2010]
24. y ''  4 y '  4 y  8 cos 2 x
26. (𝐷2 – 4𝐷 + 3)𝑦 = 𝑠𝑖𝑛3𝑥𝑐𝑜𝑠2𝑥
28. y ''  4 y '  12 y  e 2 x  3sin 2 x
29. (𝐷3 − 𝐷)𝑦 = 2𝑒 𝑥 + 4𝑐𝑜𝑠𝑥
[June 2011]
30.
d2y
dy
 3  2y  x2
2
dx
dx
31. (𝐷2 + 4)𝑦 = 𝑥 2 + 𝑒 −𝑥
[June 2015]
32.
d2y
 4 y  x 2  2 x  log 2
dx 2
33. (𝐷2 + 3𝐷 + 2)𝑦 = 4𝑐𝑜𝑠 2 𝑥
[Aug 2020]
2
35.
d y
dy
 6  25 y  e 2 x  sin x  x [June 2018]
2
dx
dx
37. ( D3  8) y  x 4  2 x  1
39.
d2y
dy
 4  13 y  e3 x cosh 2 x  3x
2
dx
dx
d3y
d2y
dy

6
 11  6 y  e x  1
3
2
dx
dx
dx
2
d y
41.
 4 y  cosh2 x  1  3 x
2
dx
y
42. '''  6 y ''  11y '  6 y  1  x  sin x
40.
[June 2018]
[June 2018]
[June 2018]
[Dec 2017]
34. 𝑦 ′′ + 2𝑦 ′ + 𝑦 = 2𝑥 + 𝑥 2
[Aug 2020]
36. ( D 2  3D  2) y  2 x 2  sin 2 x
[June 2019]
38. (2𝐷2 + 2𝐷 + 3) 𝑦 = 𝑥 2 – 2𝑥– 1
[June 2018, 2017, 2016]
[June 2015, 2014]
[Jan 2021, June 2017, 2016, 2014]
[June 2017]
10
Calculus and Differential Equations / 21MAT11
𝑑3 𝑦
𝑑2 𝑦
DEPARTMENT OF MATHEMATICS
𝑑𝑦
43. 𝑑𝑥 3 + 𝑑𝑥 3 + 4 𝑑𝑥 + 4𝑦 = 𝑥 2 − 4𝑥 − 6
[Jan 2021]
44. D2  3D  2y  1  3x  x 2
[Dec 2016, 2011, June 2016]
45. (𝐷3 − 6𝐷2 + 11𝐷 − 6)𝑦 = 𝑒 2𝑥 +2𝑥 + 𝑐𝑜𝑠2𝑥
[Dec 2016, June 2013]
46. (𝐷 − 2)2 𝑦 = 8(𝑒 2𝑥 + 𝑠𝑖𝑛2𝑥 + 𝑥 2 )
[June 2017, 2016, 2012, Dec 2016, 2011]
dy
 1 at 𝑥 = 0
47. y ''  4 y '  5 y  2 cosh x , find 𝑦 when y  0 and
[June 2018, 2017]
dx
48. y  4 y '  4 y  8x 2 given 𝑦(0) = 1, 𝑦 ′ (0) = 2
[June 2016]
Solve the following linear differential equations by the method of variation of parameters:
1. y  a 2 y  sec a x 
3. y  y  sec x tan x
[Dec2016, 2014, 2010]
[Aug 2020,June 2018]
5. y  y  cosec x
7. y  y  tan x
[June 2016]
[Jan2021, June 2013]
9. y  2y  2y  ex tanx
11. y  y 

1
1  Sinx

[June 2015]
[June 2018, 2015]
 
13. D2  3D  2 y  cos e x
2. y  y  sec x 
4. y  y  cosec x cot x
[June 2019, Dec 2017]
[Dec 2016]
-x
3
6. y  2y  2y  e sec x
[June 2018]
8. y  2 y  y  e log x
1
10. y  3y  2y 
1  e- x
x
12.
[Dec 2014]
[June 2010]
d2y
2
y
2
dx
1 ex
[June 2016]
14. (𝐷2 − 2𝐷 + 1)𝑦 = 𝑒 𝑥 𝑙𝑜𝑔𝑥
e 3x
x2
16. y  4y  tan(2x)
15. y  6y  9y 
[Dec 2016, 2011, June 2011]
[Jan 2020, Dec 2019, 2017, 2011, June 2017, 2012]
Cauchy’s and Legendre’s linear differential equations:
Solve the following differential equations:
d2y
dy
 x  y  log x [Dec 2017, June 2017, 2010]
1. x
2
dx
dx
2
3. x 3
2
d3y
dy
2 d y

3
x
 x  y  x  log x
3
2
dx
dx
dx
5. x 2 y  xy  9 y  3x 2  sin(3 log x)
7. x 3
2
d3y
dy
2 d y

3
x
 x  8 y  65 cos(log x)
3
2
dx
dx
dx
8. x 2
d2y
dy
 3x  4 y  (1  x) 2
2
dx
dx
d2y
dy
 x  9 y  sin(3 log x)  3x 2
9. x
2
dx
dx
2
d2y
dy
 4 x  6 y  cos(2 log x)
10. x
2
dx
dx
2
11. x y  5xy  8 y  2 log x
2
2
12. 3x  2 y  53x  2y  3y  x 2  x  1
13. (2 x  5) 2
[Dec 2017]
2. x D 2 y  2
y
1
 x 2
x
x
[June 2018]
4. x 2 y  xy  y  2 cos 2 (log x)
[June 2011]
d2y
dy
 5x  13 y  log x  x 2
2
dx
dx
[Dec 2016]
6. x 2
[Dec 2016]
[June 2016, Dec 2010]
[Aug 2020, Dec 2016, Dec 2011]
[June 2019]
[Jan 2020, Dec 2019]
[June 2018]
2
d y
dy
 6(2 x  5)  8 y  6 x
2
dx
dx
[Dec 2017]
11
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
14. (1  x) 2 y  (1  x) y  y  4 cos[log(1  x) ]
[June 2016]
d2y
dy
 6(2 x  3)  6 y  log( 2 x  3)
2
dx
dx
2
16. 3x  2 y  33x  2y  36y  8x 2  4x  1
15. (2 x  3) 2
[June 2015]
[June 2018, 2016, Dec 2014]
17. (1  x) 2 y  (1  x) y  y  2 sin[log(1  x)]
[June 2019, 2012, Dec 2017, 2016, 2011]
d2y
dy
 (2 x  3)  12 y  6 x
2
dx
dx
19. (1  x) 2 y  (1  x) y  y  sin[log(1  x) 2 ]
18. (2 x  3) 2
[June 2018, 2017, Dec 2016]
[June 2017]
20. (2𝑥 + 1)2 𝑦 ′′ − 6(2𝑥 + 1)𝑦 ′ + 16𝑦 = 8(2𝑥 + 1)2
[Aug 2020]
Self-Study:
Applications to oscillations of a spring and L-C-R circuits:
1. A condenser of capacity 𝐶 discharged through an inductance 𝐿 and resistance 𝑅 in series and charge q at
𝑑2 𝑞
𝑑𝑞
𝑞
time satisfying the equation 𝐿 𝑑𝑡 2 + 𝑅 𝑑𝑡 + 𝐶 = Esinpt. Given 𝐿 = 0.25𝐻, 𝑅 = 250 ohms 𝐶 = 2 × 10−6
and that when 𝑡 = 0 charge 𝑞 = 0.02 and current
𝑑𝑞
= 0. Obtain 𝑞 in time 𝑡
𝑑𝑡
𝑑2 𝑞
𝑑𝑞
𝑞
2. In an LCR circuit the charge 𝑞 on a plate of condenser is given by 𝑑𝑡 2 + 𝑅 𝑑𝑡 + 𝐶 = Esinpt . The circuit is
1
tuned to resonance so that 𝑝2 = 𝐿𝐶. If initially the current 𝐼 and the charge be zero. Show that for all small
𝑅
𝐸𝑡
values of 𝐿 , the current in the circuit at a time t is given by 2𝐿 𝑠𝑖𝑛𝑝𝑡.
3. The current 𝑖 and the charge 𝑞 in a series containing an inductance 𝐿, capacitance 𝐶, emf 𝐸, satisfy the
𝑑2 𝑞
𝑞
differential equation 𝐿 𝑑𝑡 2 + 𝐶 = E. Express 𝑞 and 𝑖 in terms of ‘𝑡’ given that 𝐿, 𝐶, 𝐸 are constants and the
value of 𝑖 and 𝑞 are both zero initially.
[June 2019]
2
4. The differential equation of a simple pendulum is
d x
2
 w0 x  F0 sin nt where w0 and F0 are constants.
2
dt
dx
 0 . Solve it.
[Dec 2019]
dt
5. The differential equation of spring fixed at upper end and a weight at its lower end is given by
Also initially x  0,
𝑑2 𝑥
𝑑𝑥
10 𝑑𝑡 2 + 𝑑𝑡 + 200𝑥 = 0. The weight is pulled down 0.25cm below the equilibrium position and then
released. Find the expression for displacement of weight from its equilibrium positionat any time 𝑡 during
its first upward motion.
6. The differential equation of spring fixed at upper end and a weight at its lower end is given by
𝑑2 𝑥
𝑑𝑡 2
𝜆𝑎
+ 𝜆2 𝑥 = λ2 𝑎𝑠𝑖𝑛𝑛𝑡. Show that if ≠ 𝜆, the displacement is given by 𝑥 = 𝜆2 −𝑛2 (𝜆𝑠𝑖𝑛𝑛𝑡 − 𝑛𝑠𝑖𝑛𝜆𝑡).
What happens when 𝑛 = 𝜆?
**********************************
12
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
MODULE V: LINEAR ALGEBRA
Rank of a Matrix:
Find the rank of the following matrices by reducing it to echelon form:
5
0 1  3  1
2 1 3
4
1 0 1



2
1
4 2 1
3


1.
[June 2018]
2.
[Dec 2017] 3. 
3 1 0
8 4 7 13 
2
2





1 1  2 0 
8 4  3  1
2
2  1 3
0 3 4
4. 
2 3 7

2 5 11
1
2
7. 
3

6
4
1
[June 2016]
5

6
2 3 0
4 3 2 
2 1 3

8 7 5
1 2 3 2 
10. 2 3 5 1
1 3 4 5
1
2
12. 
4

1
3
8 
8 12 16

2 3 4
2
4
4
6
[Jan 2021]
1 2 3 4
5. 5 6 7 8 


8 7 0 5
3 4 1
1 7 3
8. 
5 2 5

9 3 7
[Dec 2019, June 2014]
[Dec 2018, June 2011]
2
1 
4

7
[Dec 2015]
[Aug 2021]
0 2 1
1 3 4
3 4 7

3 1 4
[Dec 2017]
 1 2 0  1
6.  3 4 1 2 


 2 3 2 5 
[Dec 2013]
 2 1 3 1
1 2 3 1

9. 
1 0 1 1 


 0 1 1 1
[Aug 2020]
2 2 3 
1
2
5  4 6 

11.
[June 2019, Dec 2019, 2018, 2012]
  1  3 2  2


4 1 6 
2
2 3  1  1 
1  1  2  4 

13. 
3 1
3  2


0  7
6 3
91
 2  1  3  1
92
1


2
3

1
 [Dec 2107, 2015, June 2015] 15. 93
14. 
1
0
1
1



94
1  1  1
0
95
[June 2016, Dec 2016]
92 93 94 95
93 94 95 96
94 95 96 97 

95 96 97 98
96 97 98 99
[Dec 2010]
Consistency of System of Linear Equations:
1. For what values of  and  following simultaneous equations have i) no solution ii) a unique solution
iii) an infinite number of solutions? 𝑥 + 𝑦 + 𝑧 = 6, 𝑥 + 2𝑦 + 3𝑧 = 10,
𝑥 + 2𝑦 +  𝑧 = 
[Aug 2021, Jan 2021, June 2019, 2016, 2014, Dec 2012, 2010]
2. Investigate the value of  and  so that the equations 2x + 3y + 5z = 9; 7x + 3y - 2z = 8; 2x + y + z = 
have i) no solution ii) a unique solution iii) an infinite number of solutions?
[June 2011]
3. Find the value of  and  such that the system x  2 y  3z  6; x  3 y  5z  9; 2 x  5 y  z   have
i) no solution ii) a unique solution iii) an infinite number of solutions?
[Jan 2019, Dec 2018]
13
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
4. Show that the following system is consistent and solve: x  y  z  4; 2 x  y  z  1; x  2 y  2 z  2
5. Test for consistency and solve: 5x + 3y + 7z = 4 ; 3x + 26y + 2z = 9; 7x + 2y +10z = 5
[June 2018, Dec 2013]
[Dec 2017, 2015]
6. Test for consistency and solve: 4𝑥 − 2𝑦 + 6𝑧 = 8 , 𝑥 + 𝑦 − 3𝑧 = −1 , 15𝑥 − 3𝑦 + 9𝑧 = 21
7. Test for consistency and solve: 5𝑥 + 𝑦 + 3𝑧 = 20 ,2𝑥 + 5𝑦 + 2𝑧 = 18 , 3𝑥 + 2𝑦 + 𝑧 = 14
[Aug 2020]
8. Test for consistency and solve: x  2 y  2 z  5; 2 x  y  3z  6; 3x  y  2 z  4; and x  y  z  1 [June 2015]
Gauss-Elimination Method:
Solve the following system of equations by Gauss Elimination method:
1. 𝑥 + 2𝑦 + 𝑧 = 3, 2𝑥 + 3𝑦 + 2𝑧 = 5, 3𝑥 − 5𝑦 + 5𝑧 = 2
[June 2017]
2. 𝑥 + 𝑦 + 𝑧 = 9, 𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 3
[Aug 2021, Dec 2017, 2012]
3. 2𝑥 − 3𝑦 + 4𝑧 = 7, 5𝑥 − 2𝑦 + 2𝑧 = 7, 6𝑥 − 3𝑦 + 10𝑧 = 23
[June 2017]
4. 2𝑥 + 𝑦 + 4𝑧 = 12, 4𝑥 + 11𝑦 − 𝑧 = 33, 8𝑥 − 3𝑦 + 2𝑧 = 20
[Dec 2016, 2015]
5. 4𝑥 + 𝑦 + 𝑧 = 4, 𝑥 + 4𝑦 − 2𝑧 = 4, 3𝑥 + 2𝑦 − 4𝑧 = 6
[June 2016]
6. 3𝑥 − 𝑦 + 2𝑧 = 12, 𝑥 + 2𝑦 + 3𝑧 = 11, 2𝑥 − 2𝑦 − 𝑧 = 2
[Dec 2014]
7. 2𝑥 − 𝑦 + 3𝑧 = 1 , −3𝑥 + 4𝑦 − 5𝑧 = 0 , 𝑥 + 3𝑦 + 6𝑧 = 0
[June 2014]
8. 2𝑥 + 5𝑦 + 7𝑧 = 52 , 2𝑥 + 𝑦 − 𝑧 = 0 , 𝑥 + 𝑦 + 𝑧 = 9
[Jan 2019, Dec 2018]
9. 𝑥 − 2𝑦 + 3𝑧 = 2 , 3𝑥 − 𝑦 + 4𝑧 = 4 , 2𝑥 + 𝑦 − 2𝑧 = 5
[Dec 2019]
10. 5𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 4 , 𝑥1 + 7𝑥2 + 𝑥3 + 𝑥4 = 12 , 𝑥1 + 𝑥2 + 6𝑥3 + 𝑥4 = −5 , 𝑥1 + 𝑥2 + 𝑥3 + 4𝑥4 = −6
[June 2015]
Gauss-Jordan Method:
Solve the following system of equations by Gauss Jordan method:
1. 𝑥 + 𝑦 + 𝑧 = 9, 𝑥 − 2𝑦 + 3𝑧 = 8, 2𝑥 + 𝑦 − 𝑧 = 3 [Jan 2021, June 2018, 2017, 2016, 2015, Dec 2017, 2010]
2. 𝑥 + 2𝑦 + 𝑧 = 3, 2𝑥 + 3𝑦 + 3𝑧 = 10, 3𝑥 − 𝑦 + 2𝑧 = 13
[Dec 2017]
3. 2𝑥 + 𝑦 + 𝑧 = 10, 3𝑥 + 2𝑦 + 3𝑧 = 18, 𝑥 + 4𝑦 + 9𝑧 = 16
[Dec 2016]
4. 𝑥 + 𝑦 + 𝑧 = 8, −𝑥 − 𝑦 + 2𝑧 = −4, 3𝑥 + 5𝑦 − 7𝑧 = 14
[Dec 2015]
5. 2𝑥 − 3𝑦 + 𝑧 = 1, 𝑥 + 4𝑦 + 5𝑧 = 25, 3𝑥 − 4𝑦 + 𝑧 = 2
[Dec 2017]
6. 𝑥 + 𝑦 + 𝑧 = 1, 4𝑥 + 3𝑦 − 𝑧 = 6, 3𝑥 + 5𝑦 + 3𝑧 = 4
[Dec 2016]
7. 2𝑥 + 5𝑦 + 7𝑧 = 52 , 2𝑥 + 𝑦 − 𝑧 = 0 , 𝑥 + 𝑦 + 𝑧 = 9
[Aug 2020, June 2019, Dec 2015, 2013, 2012]
8. 2𝑥1 + 𝑥2 + 3𝑥3 = 1, 4𝑥1 + 4𝑥2 + 7𝑥3 = 1, 2𝑥1 + 5𝑥2 + 9𝑥3 = 3
[Dec 2019]
9. 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 2 , 2𝑥1 − 𝑥2 + 2𝑥3 − 𝑥4 = −5 , 3𝑥1 + 2𝑥2 + 3𝑥3 + 4𝑥4 = 7 , 𝑥1 − 2𝑥2 − 3𝑥3 + 2𝑥4 = 5
Gauss-Seidel Method:
Solve the following system of equations by Gauss Seidel Method performing 3 iterations:
1. 20𝑥 + 𝑦 − 2𝑧 = 17 , 3𝑥 + 20𝑦 − 𝑧 = −18 , 2𝑥 − 3𝑦 + 20𝑧 = 25
[Aug 2021, June 2019, Dec 2019, 2018, 2017, Dec 2017, 2016, 2015, 2012]
2. 3𝑥 + 8𝑦 + 29𝑧 = 71, 83𝑥 + 11𝑦 − 4𝑧 = 95; 7𝑥 + 52𝑦 + 13𝑧 = 104
[Jan 2021, June 2018]
3. 10𝑥 + 2𝑦 + 𝑧 = 9 , 𝑥 + 10𝑦 − 𝑧 = −22 , −2𝑥 + 3𝑦 + 10𝑧 = 22
[June 2017]
4. 𝑥 + 𝑦 + 54𝑧 = 110 , 27𝑥 + 6𝑦 − 𝑧 = 85 , 6𝑥 + 15𝑦 + 2𝑧 = 72
[Dec 2015, 2014]
9 4 6
5. 5𝑥 − 𝑦 = 9, 𝑥 − 5𝑦 + 𝑧 = −4, 𝑦 − 5𝑧 = 6 taking (5 , 5 , 5) as first approximation.
6.
7.
8.
9.
[Dec 2011]
10𝑥 + 𝑦 + 𝑧 = 12, 𝑥 + 10𝑦 + 𝑧 = 12, 𝑥 + 𝑦 + 10𝑧 = 12
[June 2016]
12𝑥 + 𝑦 + 𝑧 = 31, 2𝑥 + 8𝑦 − 𝑧 = 24, 3𝑥 + 4𝑦 + 10𝑧 = 58
[Jan 2019, Dec 2018]
5𝑥 + 2𝑦 + 𝑧 = 12 , 𝑥 + 4𝑦 + 2𝑧 = 15 , 𝑥 + 2𝑦 + 5𝑧 = 20
[Aug 2020]
5𝑥 + 2𝑦 + 𝑧 = 12 , 𝑥 + 4𝑦 + 2𝑧 = 15 , 𝑥 + 2𝑦 + 5𝑧 = 20 taking initial approximation as (1,0,3)
14
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
Rayleigh Power Method:
Find the largest Eigen value and the corresponding Eigen vector of the following matrices using power method:
2 0 1 
1. 𝐴 = 0 2 0 . Take (1,0,0) as initial vector. Carry out six iterations.


1 0 2
[June 2019, Dec 2019, 2018, 2016, Dec 2017, 2016, 2015]
 6 2 2 
2. A   2 3  1 . Take (1, 1, 1) as initial vector. Carry out five iterations.


 2  1 3 
[Aug 2021, Aug 2020, Jan 2019, Dec 2018, 2017, 2011, June 2016, 2015, 2012]
3.
4.
 2 1 0 
A   1 2  1 . Take (1, 0, 0) as initial vector. Carry out four iterations.
 0  1 2 
[June 2018, 2012, Dec 2016, 2015, 2013]
1 6 1
[June 2018, 2017]
A  1 2 0 . Take (1, 1, 0) as initial vector. Carry out six iterations.
0 0 3
5.
 1 3  1
A   3 2 4  . Take (0, 0, 1) as initial vector. Carry out five iterations.
 1 4 10 
[Dec 2014]
6.
5 0 1 
A  0  2 0 . Take (1, 0, 0) as initial vector. Carry out six iterations.
1 0 5
[Dec 2017]
7.
 4 1  1
A   2 3  1 . Take (1, 0, 0) as initial vector. Carry out seven iterations.
 2 1 5 
[Jan 2021, Dec 2014]
8.
25 1 2 
A   1 3 0  . Take (1, 0, 0) as initial vector. Carry out seven iterations.
 2 0  4
[June 2013]
9.
10 2 1 
A   2 10 1  . Take (1, 1, 0) as initial vector. Carry out six iterations.
 2 1 10
[June 2014]
Self-study:
Gauss-Jacobi iterative Method:
1.
2.
3.
4.
5.
28𝑥 + 4𝑦 − 𝑧 = 32, 2𝑥 + 17𝑦 + 4𝑧 = 35, 𝑥 + 3𝑦 + 10𝑧 = 24
10𝑥 + 2𝑦 + 𝑧 = 9 , 2𝑥 + 20𝑦 − 2𝑧 = −44 , −2𝑥 + 3𝑦 + 10𝑧 = 22
6𝑥 − 𝑦 − 𝑧 = 19 , 3𝑥 + 4𝑦 + 𝑧 = 26 , 𝑥 + 2𝑦 + 6𝑧 = 22
3𝑥 + 𝑦 + 𝑧 = 1 , 𝑥 + 3𝑦 − 𝑧 = 11 , 𝑥 − 2𝑦 + 4𝑧 = 21
23𝑥 + 13𝑦 + 3𝑧 = 29 , 5𝑥 + 23𝑦 + 7𝑧 = 37 , 11𝑥 + 𝑦 + 23𝑧 = 43
15
Calculus and Differential Equations / 21MAT11
DEPARTMENT OF MATHEMATICS
Cayley-Hamilton Method:
1. By using Cayley-Hamilton theorem, find the inverse of the following matrices:
1 0 2
(i) A   0 2 1 


 2 0 3 
 2 1 1 
(ii) A   1 2 1


 1 1 2 
1 0 2
(iii) A   0 2 1 


 2 0 2 
*************************************
16