Important Questions for VTU Exam M2

Engineering Mathematics 2

CMRIT

10MAT21

UNIT 1 DIFFERENTIAL EQUATION 1 Problems on Solvable for P 2

 dy  1. Solve y    dx 

+ ( x − y )

dy

− x =

dx

0

2. Solve x ( y ′) − (2 x + 3 y ) y ′ + 6 y = 0 2

 dy  3. Solve    dx 

2

 dy  4. Solve xy    dx 

−7 2

−

dy dx

( x

_ 12 = 0

2

− y

2

) dy + xy = 0 dx

5. Solve p 2 − 2 p sinh x − 1 = 0 6. Solve p 2 + 2 py cot x = y 2

(5Marks -June 2012)

7. Solve xp 2 + xp − ( y 2 + y ) = 0 8. Solve p( p + y ) = x( x + y ) 9. Solve

dy dx

−

dx dy

=

x y

−

(4Marks -July 2011,Dec 2011)

y x

10. Solve p 2 + 2 p cosh x + 1 = 0

(4Marks -Jan 2014)

11. Solve p 2 + p ( x + y ) + xy = 0

(4Marks -Jan 2013)

Problems on Solvable for x or y 12. Solve y − 2 px = tan −1 (xp 2 ) 13. Obtain the general solution and singular solution of y + px = p 2 x 4 14. Find the general solution and singular solution of x 2 p 4 + 2 xp − y = 0 15. Solve y

=

p sin p + cos p

16. Obtain the general solution and singular solution of p 2 + 4 x 5 p − 12 x 4 y = 0 (5Marks -June 2012) 17. Obtain the general solution and singular solution of xp 2 − 2 yp + ax = 0 (6Marks -Jan 2013) 18. Solve p 3 − 4 xyp + 8 y 2 = 0

(5Marks -July 2011)

19. Solve y = 2 px + p 2 y

(6Marks -Dec 2011)

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Department of Mathematics


CMRIT

10MAT21

Problems on Clairaut’s equation ( y = px + f ( p ) form) 20. Solve y = px +

a p

21. Solve p = log( px − y ) 22. Solve ( p + 1) ( y − px ) = 1 2

23. Solve xp 2 + px − py + 1 − y = 0 , Also find the singular solution. 24. Solve xp 2 − py + kp + a = 0 , Also find the singular solution. 25. Obtain the general solution and singular solution of xp 3 − yp 2 + 1 = 0 (6Marks -Dec 2011) 26. Obtain the general solution and singular solution of sin px cos y = cos px sin y + p (6Marks-Jan 2013) 27. Solve the equation ( px − y )( py + x ) = 2 p using the substitution X = x 2 , Y = y 2 (6Marks -Dec 2011) 28. Solve the equation ( px − y )( py + x ) = a 2 p using the substitution X = x 2 , Y = y 2 29. Find the general solution of x 2 ( y − px ) = p 2 y using the substitution X = x 2 , Y = y 2 . Also find the singular solution. 30. Find the general solution of y 2 ( y − px ) = x 4 p 2 using the substitution X =

1 x

, Y =

1 y

. Also find

the singular solution.

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CMRIT

10MAT21

UNIT 2 DIFFERENTIAL EQUATION 2 Problems to find Complementary function y c 2

1. Solve

d y dx

2

+3

dy

2

2. Solve

d y dx

2

3. Solve

dx

2

dy

−7

2

d y

−6

+ 2y =

dx

0

+ 12 y =

dx dy

0

+ 9y = 0

dx

4. Solve ( D 2 − 3 D + 2)y = 0 5. Solve

6. Solve

7. Solve

d 3 y dx

3

d 3 y dx

3

d 3 y dx

3

− 13

−6

−3

dy

+ 12 y =

dx

d 2 y dx

d 2 y dx

+ 11

2

2

−3

0

dy

− 6y =

dx

dy dx

−

0

y=0

(4Marks -Jan 2014)

8. Solve ( D 4 + 4 D 3 − 5 D 2 − 36 D − 36 )y = 0

(4Marks -Dec 2011)

9. Solve ( D 3 + D 2 + 4 D + 4 )y = 0

(4Marks -Jun 2010)

10. Solve

d 3 y dx 3

− 8y =

0

11. Solve ( D 4 − 2 D 3 + 2 D 2 − 2 D + 1)y = 0 12. Solve ( D 4 − 8 D 2 + 16 )y = 0 13. Solve ( D 2 + 1) ( D − 1) y = 0 2

2

Problems to find Particular Integral (PI or y p) of the type φ ( x ) = e a x 14. Solve y ′′ − 6 y ′ + 9 y = 5e −2 x 15. Solve y ′′ + 3 y ′ + 2 y = cosh x 16. Solve y ′′ − 8 y ′ + 16 y = 3e 4 x 17. Solve x ′′′(t ) − 8 x(t ) = (1 − e t )

2

3

18. Solve

d y dx

3

2

+3

d y dx

2

+3

dy dx

+ y =

5e 2 x + 6e − x + 7

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CMRIT

10MAT21

Problems to find Particular Integral (PI or y p) of the type φ ( x ) = sin ax or cos ax 19. Solve y ′′ − 4 y ′ + 13 y = cos 2 x 20. Solve y ′′ − 3 y ′ + 2 y = 2 sin x cos x

(6Marks -JAN 2014)

21. Solve y ′′′ − 3 y ′′ + 9 y ′ − 27 y = cos 3 x 22. Solve

d 2 y dx

+3

2

dy dx

+

2 2 y = 4 cos x

23. Solve ( D 2 + 4 ) y = sin 2 x 24. Solve y ′′ + 9 y = cos 2 x cos x 25. Solve ( D 2 − 4 D + 3) y = sin 3 x cos 2 x 26. Solve ( D 3 − 1) y = 3 cos 2 x

(6Marks -DEC 2010)

27. Solve ( D 3 + 4 D ) y = sin 2 x

(4Marks - JAN 2010)

Problems to find Particular Integral (PI or y p) of the type φ ( x ) = ax n

+ bx

n −1

+−−−

28. Solve y ′′ + 3 y ′ + 2 y = 12 x 2 29. Solve ( D 3 + 8) y = x 4 + 2 x + 1 30. Solve ( D 2 − 5 D + 1) y = 1 + x 2

(4Marks – JULY 2009)

31. Solve ( D 2 + 3 D + 2 ) y = 1 + 3 x + x 2 32. Solve

33. Solve

d 3 y dx

3

+

d 3 y dx

3

2

d 2 y dx

2

+

− 8 y = x

dy dx

(x

2

=

x3

)

+1

Problems to find Particular Integral (PI or y p) on all types 34. Solve

d 2 y dx

2

−4

dy dx

+ 4 y =

e 2 x + cos x + 4

(6Marks – JAN 2013)

35. Solve ( D 3 − D ) y = 2e x + 4 cos x


36. Solve ( D − 2 ) y = 8(e 2 x + sin 2 x ) 2

(4Marks – DEC 2011,JUNE 2012)

x 37. Solve ( D 3 + D 2 + 4 D + 4 ) y = 3e − − 4 x − 6

38. Solve ( D 3 + 2 D 2 + D ) y = e − x + sin 2 x

(6Marks – DEC 2011)

Problems to find Particular Integral (PI or y p) of the type e ax φ ( x ) �



CMRIT

10MAT21

39. Solve ( D 2 − 2 D + 5) y = e 2 x sin x


40. Solve ( D 2 − 4 D + 3) y = 2 xe 3 x 41. Solve y ′′ + 2 y ′ + 5 y = e − x sin 2 x 42. Solve y ′′ + 4 y ′ + 3 y = e −3 x x 2 43. Solve ( D 2 + 2 ) y = x 2 e 3 x + cos 2 x


x 44. Solve ( D 2 − 2 D − 3) y = e cos x


Problems to find Particular Integral (PI or y p) of the type x sin ax or x cos ax 45. Solve y ′′ + 16 y = x sin 3 x

(6Marks – JUNE 2010)

46. Solve ( D 2 − 1) y = x cos x 47. Solve y ′′ − y = x 2 cox3 x 48. Solve y ′′ − 2 y ′ + y = x cos x

(6Marks– DEC 2011)

x 49. Solve y ′′ − 2 y ′ + y = x e cos x


Problems on Simultaneous Differential Equations 50. Solve 51. Solve 52. Solve

dx dt dx dt dx dt

+ y =

+

t e ,

dy dt

− x =

dy

2 y = − sin t ,

− 2 y =

dt dy

cos 2t ,

dt

e−

t

− 2 x =

cos t

+ 2 x =

sin 2t given x = 1, y = 0 at t = 0 (6Marks – JUNE2012, JAN 2014)

53. Solve 54. Solve 55. Solve

dx dt dx dt dx dt

=

2 x − 3 y ,

+ 2 x − 2 y =

− 7 x + y =

dy dt t ,

0 ,

= y − 2 x

dy dt dy dt

+ 2 x +

(6Marks – JAN 2013) y=0

− 2 x − 5 y =

0

(6Marks – JULY 2011) (6Marks – DEC 2011)

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CMRIT

10MAT21

UNIT 3 DIFFERENTIAL EQUATION 3 Solve the following differential equation using Variation of parameters. 2

1. Solve

d y dx

2. Solve

2

d 2 y dx

2

+ y =

cos ecx

+ y =

tan x

(4Marks – JULY 2013, 2009)

2 3. Solve D + 4 y = cos ecx 2

4. Solve

d y dx

2

+ y =

sec x tan x

5. Solve y ′′ − 2 y ′ + y =

e x


x

6. Solve y ′′ − 3 y ′ + 2 y =

1


1 + e − x

7. Solve y ′′ − 2 y ′ + 2 y = e x tan x


8. Solve y ′′ − 2 y ′ + y = e x log x 2

9. Solve

d y dx

2

−6

dy dx

+ 9 y =

e

3x

x

(4Marks – JULY 2011, Dec 2011)

2

10. Solve y ′′ + 4 y = tan 2 x

(5Marks – JUNE 2012, DEC 2011)

11. Solve y ′′ + a 2 y = sec ax

(5Marks – JAN 2014, DEC 2010)

Solve the following Cauchy’s Linear Differential equations 2 dy 2 d y 12. Solve x − 3 x + 4 y = 2 dx

(1 + x )2


dx

3

13. Solve x

d y dx

3

2

+

d y dx

2

=

1 x

2 dy 2 d y x 14. Solve x + 4 x + 2 y = e 2 dx


dx

2

15. Solve x

d y dx

2

−

2 y x

= x +

1 x

2 �



CMRIT

2 dy 2 d y + 4 x − y = 16. Solve 4 x 2 dx

10MAT21

4x 2

dx

2 dy 2 d y 17. Solve x − x + y = 2 dx

log x


dx

2

18. Solve x y ′′ + x y ′ + y = 2 cos

2

(log x )

3 2 dy 3 d y 2 d y 19. Solve x + 3 x − 2 x + 2y = 3 2 dx

dx


0

dx

2

20. Solve x y ′′ − x y ′ + 2 y = x sin (log x ) 2 dy 2 d y 21. Solve x − 4 x + 6 y = 2 dx

(6Marks – JAN 2013,JULY 2014)

cos(2 log x )


dx

2 dy 2 d y 2 22. Solve x + x + 9 y = 3 x + sin (3 log x ) 2 dx


dx

Solve the following Legendre,s linear Differential equation 2 dy 2 d y 23. Solve (1 + x ) + (1 + x ) + y = sin 2[log(1 + x)] 2 dx


dx

2

24. Solve (1 + x ) y ′′ + (1 + x ) y ′ + y = 2 sin [log(1 + x)] 2

25. Solve (2 x + 1) y ′′ − 6(2 x + 1) y ′ + 16 y = 8(2 x + 1)

(6Marks – JUNE 2012,JULY 2009)

2

2

26. Solve ( x + 2) y ′′ − ( x + 2 ) y ′ + y = 3 x + 4 2

2 27. Solve (3 x + 2 ) y ′′ + 3(3 x + 2 ) y ′ − 36 y = 8 x + 4 x + 1 2

28. Solve (3 x − 2 ) y ′′ − 3(3 x − 2) y ′ = 9(3 x − 2) sin [log(3 x − 2)] 2 dy 2 d y 29. Solve (2 x + 3) − (2 x + 3) − 12 y = 2 dx

6x

dx

Problems on Series Solution 30. Obtain the Series Solution of the equation

dy dx

− 2 x

y = 0 (6Marks – DEC JAN 2014)

31. Obtain the Power Series Solution of the equation y ′′ + y = 0 32. Solve

d 2 y dx 2

+ x

y = 0 by obtaining the solution in the form of series.

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Engineering Mathematics 2 33. Solve

d 2 y dx 2

− y =

CMRIT

10MAT21

0 by obtaining the power series solution.

34. Solve (1 + x 2 ) y ′′ + x y ′ − y = 0 in series solution

(6Marks –JUNE-JULY 2013)

Problems on Fobenious Method or Generalized Power Series 2

35. Solve by Frobenius method 4 x

d y dx

2

+2

dy dx

2

36. Solve by Frobenius method 2 x

d y dx

2

+3

+

dy dx

y=0

−

y=0

(6Marks DEC 2011,JUNE 2012)

(6Marks JULY 2011) 2

37. Obtain the Frobenius type series solution of the equation x

d y dx

2

+ y =

0 (6Marks JAN 2013)

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CMRIT

10MAT21

UNIT 7 LAPLACE TRANSFORM Find the Laplace Transform of the following functions 1. Find L sin 2 t 2. Find L sin 3 2t

(6M JAN 2014)

3. Find L cos 2 t 4. Find L cosh 2 t 5. Find L e −2t sinh 4t 6. Find L 2 t + cos 3t

(4M JULY 2009)

7. Find L[sin 5t cos 2t ] 8. Find L[sin t sin 2t sin 3t ]

(4M JAN 2013)

9. Find L[cos t cos 2t cos 3t ] 3

10. Find L (3t + 4 ) + 5 t 11. Find L e at + 2t n − 3 sin 3t + 4 cosh 2t

(4M JUNE 2010)

12. Find L[cos(2t + 3)] Problems on Property L e at f (t ) 13. Find L e −2 t (2 cos 5t − sin 5t ) 14. Find L t 3 cosh t 15. Find L e −3t sin 5t sin 3t

(4M DEC 2011)

16. Find L e 3t sin 5t sin 3t

(4M DEC 2010)

17. Find L t 3 e 2t 18. Find L t 2 e 2 t 19. Find L e −t cos 2 3t

(4M JAN 2010)

20. Find L[cosh 2t cos 2t ] Problems on Property L t n f (t ) 21. Find L[t cos at ] �



CMRIT

10MAT21

22. Find L t 2 sin at 23. Find L t (sin 3 t − cos 3 t )

(5M JULY 2011)

24. Find L[t sin 3t cos t ] 25. Find L t sin 2 t ∞

26. Find the value of

∫ t e 3

−t

sin t dt using Laplace transforms

(5M DEC 2011, JUNE 2012)

0 ∞


∫ te

− 2t

sin 4t dt using Laplace transforms

0

 f (t )  Problems onProperty L   t 

1 − e at  28. Find L    t   cos at − cos bt   t 

29. Find L 

 sinh t   t 

30. Find L 

 sin 2 t  31. Find L    t   e −t sin t  32. Find L   and hence find t   ∞


∫

e


∫

−e

∫ 0

t

dt

(5M JUJY 2013, JULY 2009)

dt using Laplace transforms

cos 6t − cos 4t

0

e − t sin t

− bt

t

0 ∞

− at

∞

t

dt using Laplace transforms

 t  Problems on Property L ∫ f (t ) dt  0   t



0



∫

35. Find L  sinh at sin at dt 

 t  36. Find L  ∫ t cos at dt  0  ��



CMRIT

10MAT21

 t sin at  37. Find L  ∫ dt  t 0  Problems on Periodic function.

38. Prove that L[ f (t ) ] =

T

1 1− e

− sT

∫e

− st

f (t ) dt

(6M JUNE 2010)

0

39. If f (t ) = t 2 , 0 < t < 2 and f (t + 2) = f (t ) find L[ f (t ) ]

 E  40. If f (t ) =  − E

0 < t < a a

2

<

2

t < a

, where f (t + a ) = f (t ) then show that L[ f (t ) ] =

E S

 as    4 

tanh 

(6M DEC 2010,JUNE 2011)

 t 41. If f (t ) =  2 a − t

0 ≤ t < a a ≤ t < 2a

, where f (t + 2a ) = f (t ) then show that L[ f (t ) ] =

1 2

S

 as    2 

tanh 

(6M DEC 2011)

 t 42. Find the value of f (t ) =  π − t

0 ≤ t < π

π ≤ t < 2π

, where f (t + 2π ) = f (t ) (6M JUNE 2012)

 E sin ω t 0 ≤ t ≤ π  ω find L[ f (t ) ] 43. A periodic function of period 2π is defined by f (t ) =  ω π 2 π t t π − ≤ ≤  ω ω (6M JAN 2013) 44. Find the Laplace transform of the square wave function of period 2a defined by

 k

0 < t < a

− k

a < t < 2a

f (t ) = 

(6M JULY 2009) Problems on Unit step function. 45. Find L (e t −1 + sin (t − 1)) u (t − 1)

(6M DEC 2010)

46. Find L[sin t u(t − 1)] 47. Find L (3t 2 + 4t + 5) u (t − 3) 48. Find L (t 3 + t 2 + t + 1) u (t + 1) 49. Find L (1 − e 2t ) u (t + 1) 50. Find L e − t u (t − 2) ��



CMRIT

10MAT21

Express the following functions in terms of Heaviside or Unit Step Function and hence find their Laplace Transforms.

t 51. f (t ) =  5

0 < t < 4 t > 4

sin 2t 52. f (t ) =  0

0 < t < π t > π

sin t  53. f (t ) =  cos t

0 < t < π

2 t > π 2

2t 54. f (t ) =  1

0 < t < π

t 2 55. f (t ) =  4

0 < t ≤ 3

(6M JUNE 2010)

(6M JAN 2013)

t > π

(6M DEC 2011)

t > 3

1 0 < t ≤ 1  56. f (t ) = t 1 < t ≤ 2 t 2 t > 2 

cos t  57. f (t ) = 1 sin t 

(6M DEC 2011)

0 < t ≤ π π < t ≤ 2π t > 2π

cos t 0 < t ≤ π  58. f (t ) = cos 2t π < t ≤ 2π cos 3t t > 2π  sin t 0 < t ≤ π  59. f (t ) = sin 2t π < t ≤ 2π sin 3t t > 2π  t 0 < t ≤ 2  60. f (t ) = 4t 2 < t ≤ 4 8 t > 4  t − 1  61. f (t ) = − t − 3  0 

(6M JAN 2014, JAN 2010, JULY 2011)

1 < t ≤ 2 2 < t ≤ 3

(6M JULY 2013)

otherwise

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CMRIT

10MAT21

UNIT 8 INVERSE LAPLACE TRANSFORM ---- Will upload shortly

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Important Questions for VTU Exam M2

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