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)

�

Department of Mathematics

Engineering Mathematics 2

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.

�

Department of Mathematics

Engineering Mathematics 2

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

�

Department of Mathematics

Engineering Mathematics 2

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

(5Marks – JULY 2011)

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 ) �

Department of Mathematics

Engineering Mathematics 2

CMRIT

10MAT21

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

(6Marks – JULY 2009)

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

(5Marks – JULY 2011)

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

(6Marks – JAN 2010)

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

(4Marks – JAN 2013)

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)

�

Department of Mathematics

Engineering Mathematics 2

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

(4Marks – JAN 2013)

 x

6. Solve  y ′′ − 3 y ′ + 2 y =

1

(6Marks – JUNE 2010)

1 + e − x

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

(6Marks – JAN 2010)

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

(4Marks – JULY 2010)

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

(5Marks – JAN 2014)

dx

2

15. Solve  x

d   y dx

2

−

2 y  x

=  x +

1  x

2 �

Department of Mathematics

Engineering Mathematics 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

(4Marks – JUNE 2010)

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

(5Marks – JULY 2011)

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 )

(4Marks – JAN 2010)

dx

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

(6Marks – DEC 2011)

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

(6Marks – DEC 2011)

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.

�

Department of Mathematics

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)

�

Department of Mathematics

Engineering Mathematics 2

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 ] �

Department of Mathematics

Engineering Mathematics 2

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 ∞

27. Find the value of

∫ 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    ∞

33. Find the value of

∫

e

34. Find the value of

∫

−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  ��

Department of Mathematics

Engineering Mathematics 2

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) ��

Department of Mathematics

Engineering Mathematics 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

��

Department of Mathematics

Engineering Mathematics 2

CMRIT

10MAT21

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

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Department of Mathematics