K. J.

Somaiya College of Engineering, Mumbai-77

(A Constituent college of Somaiya Vidyavihar University)

F.Y. B. Tech SEM-II

Applied Mathematics-II

Practice Problems
Differential Equation of First Order and First Degree
(Module 1: Sub-module 1.1 & 1.2)

Solve the following.

1. ( )

2. ( )

3. ( ) ( )
4. ( )
5. ( )
6. ( )

7. ( )

8. ( )

9.

10. ( ) ( )
( )
11.

12. ( ) ( )

13. ( √ ) ( √ )

14. ( ) ( )
15. ( ) ( )
16. ( )
17. ( )
18. ( ) ( )
19. ( ) ( )
20. ( ) ( )
21. ( ) ( )
22. ( )

23. ( ) Answer: * +

24. ( ) ( ) Answer: * +

25. ( ) Answer: [ ]

Module 1 Practice Problems Dr. Rachana Desai 1

 K. J. Somaiya College of Engineering, Mumbai-77
(A Constituent college of Somaiya Vidyavihar University)

26. Answer: [ ( )( )]

27. ( ) ( ) Answer:[ √ ]

28. Answer: [ ]

29. ( ) ( ) Answer: * +

30. ( ) ( ) Answer: [ ]

31. Answer: [ ( ) ]

32. Answer: [ ]

33. Answer: * +

34. Answer:* ( ) +

35. ( )( ) Answer:* +

36. Answer:* ( ) +

37. Answer: [ ( ) ]

38. Answer: [ ]

39. Answer: * +

40. Answer:* ( ) +

41. ( )( ) Answer:* +

42. Answer:* ( ) +

43. ( ) Answer:0 ( ) . / 1

44. Answer: * +

45. Answer: * ( ) +

( )
46. Answer: [ ( ) ]

47. Answer:

48. ( ) Answer: [ [ √ ] ]

Module 1 Practice Problems Dr. Rachana Desai 2

 K. J. Somaiya College of Engineering, Mumbai-77
(A Constituent college of Somaiya Vidyavihar University)

49. In a circuit containing inductance L, resistance R, and voltage E, the current I is given by
di
L + Ri = E . Find the current i at time t if at t = 0, i = 0 and L, R, E are constants.
dt
[Ans: i =
E
R

1  e  Rt / L ] 
50. A constant e. m. f. E volts is applied to a circuit containing a constant resistance R ohms in series
di
and a constant inductance L henries. The current i at any time t is given by L + Ri = E . If
dt
the initial current is zero. Show that the current builds up to half its theoretical maximum value
L
in t= log2 .
R
di
51. The equation of an L-R circuit is given by L + Ri = 10sint .If i = 0 at t = 0 express i as a
dt
function of t. [Ans: i=
10
sint  φ+ e  Rt / L

sinφ where tanφ =
L
R
]
R 2 + L2
52. The change q on the plate of a condenser of capacity C charged through a resistance R by a
dq q
steady voltage V satisfies the differential equation R + = V . If q = 0 at t = 0 , show
dt c
that  
q = CV 1  e t / RC Find also the current flowing into the plate.
V t / RC
[Ans: i = e ]
R
dQ Q
53. The differential equation of an electrical circuit is R + = V . If R=20 ohms , C=0.01 farad
dt C
5 t 5t
and V= 20 e and if Q=0 when t=0, find Q in terms of t. [Ans: Q e = t ]

54. A resistance of 100 ohms and inductance of 0.5 henries are connected in series with a battery of
di
20 volts. Find the current at any instant if the relation between L, R, E is L + Ri = E .
dt
[Ans: 
i = 0.2 1  e 200 t ] 
di
55. In a circuit of resistance R, self inductance L, the current I is given by L  
RiEcos
pt
,
dt
when E and p are constants. Find the current i at time t.

Module 1 Practice Problems Dr. Rachana Desai 3

 K. J. Somaiya College of Engineering, Mumbai-77
(A Constituent college of Somaiya Vidyavihar University)

[Ans: i = 2
E
2 2
Lpsinpt  Rcospt+ k e R t / L ]
L p +R

di
56. In a circuit of resistance R, self inductance L, the current i is given by L + Ri = Ecospt , Find
dt
the current i at any time t if i=0 at t=0.

[Ans: i=
E
sin pt + φ  e R t / L

cosφ where φ = Lp / R ]
L2 p 2 + R 2

Practice Problems
Linear Differential Equation with Constant Coefficient
(Module 1: Sub-module 1.3 & 1.4)

Solve the following.

[Ans: y  c1  c2 x e
d3y d2y dy
1. 3
 5 2
 8  4y  0
2x
 c3 e x ]
dx dx dx
d3y d2y dy
[Ans: y  c1e  c2 e  c3e3 x ]
x 2x
2. 3
 6 2
 11  6 y  0
dx dx dx
d4y
3.
4
 k4y  0
dx
[Ans:
     
y  e k / 2 x c1 cos k / 2 x  c2 sin k / 2 x  e  k / 2 x c3 cos k / 2 x  c4 sin k / 2 x ]    
d4y d2y
4. 6  9y  0 Ans: y  c1  c2 x  cos 3x  c3  c4 x sin 3x ]
d x4 d x2
d4y d2y
5. 2  y0
d x4 d x2
d4y
6.  y0
d x4
[      
y  e x / 2  c1 cos x / 2  c2 sin x / 2  e x / 2  c3 cos x / 2  c4 sin x / 2     ]
7. D 3
 D 2  D 1 y  0 
8. D 3

 3D 2  4 y  0
9. D 2

 1 D  1 y  0
2

10. D  8D 2  16 y  0  [Ans: y  c1  c2 x  cos 2 x  c3  c4 x sin 2 x ]

4


11. D 4  4 D 3  8D 2  8D  4 y  0  [Ans: y  e
x
c1  c2 xcos x  c3  c4 xsin x ]

Module 1 Practice Problems Dr. Rachana Desai 4

 K. J. Somaiya College of Engineering, Mumbai-77
(A Constituent college of Somaiya Vidyavihar University)

 
12. D 2  2 D  1 y  0
13. Solve ( )
14. Solve
15. Solve ( )
16. Solve ( )

17. Solve

18. Solve ( )
19. Solve ( )
20. Solve
21. Solve ( )
22. Solve ( )

23. Solve ( )

24. Solve ( )

25. Solve ( )

26. Solve

27. Solve ( )

Answer: * +

28. Solve ( )

Answer: * , -+

29. Solve ( )

Answer: * ( ) ( ) ( ) +

30. Solve ( )
31. Solve ( )

Answer: * ( )+

32. Solve ([ ])

Answer: * ( ) ( ) ( )+

33. Solve ( )

Answer:* +

Hint * +

34. Solve ( )
Module 1 Practice Problems Dr. Rachana Desai 5
 K. J. Somaiya College of Engineering, Mumbai-77
(A Constituent college of Somaiya Vidyavihar University)

Answer: * ( ) +

35. Solve ( )

Answer: [ ( ) ]
36. Solve ( )

37. Solve

38. Solve ( ) ( )

39. Solve ( ) ( )
40. Solve ( )

Answer: [ ( ) [ ]]

Hint: ( )( )

41. Solve

Answer: * [ ] +

42. Solve ( ) Answer: [ ( ) [ ]

43. Solve

Answer:* ( √ √ ) ( )+

44. Solve ( )

Answer:* ( ) ( )+

45. Solve ( ) [Answer : * ( ) ( )+]

Module 1 Practice Problems Dr. Rachana Desai 6