1 ) Which of the following equation represents Airy's equation ? 𝑦″ + 𝑥𝑦 = 0 , 𝑦″ − 𝑥𝑦 = 0 .

1 )   Which of the following equation represents " Airy's equation "  ?

A )   𝑦″ + 𝑥𝑦= 0  ,

B )   𝑦″ − 𝑥𝑦= 0  ,

C )   𝑦″ + (𝛼^2)𝑥𝑦= 0  ,

D )   All of above  ,

Correct answers : -   D )   All of above  ,

 

2 )  What is an integrating factor of the following equation ?

𝑥 𝑦 𝑑𝑥 + ( 2𝑥2 + 3𝑦2 – 20 ) 𝑑𝑦 = 0 ,

A )   2 / y  ,

B )   −1 / x  ,

C )   1 / 𝑦  ,

D )   1 / 𝑥𝑦  ,

Correct answers : -   C )   1 / 𝑦  ,

 

3 )  If the following function is a solution of the following equation, then the other solution is . . . . . . . ,

𝑦1 = 𝑥^2     ,    (𝑥^2) 𝑦″ − 3 𝑥 𝑦′ + 4𝑦 = 0 .

A )  ln 𝑥2   ,

B )  (𝑥^2) ln𝑥   ,

C )  ln𝑥  ,

D )  𝑥 ln𝑥

Correct answers : -    B )  (𝑥^2) ln𝑥   ,

 

4 )  In which of the the following intervals , both power series solutions centered at  x = 0 of the following equation converge ?

𝑦″ + ( cos𝑥 ) 𝑦 = 0  .

A )   ( −∞ , ∞ ) ( −∞ , ∞ )  .

B )   ( −1 , ∞ ) ( −1 , ∞ ) .

C )   [ −1/2 , 1/2 ] [ −12 , 12 ] .

D )   [ −1 , 1 ] [ −1 , 1 ] .

Correct answers : -    A )   ( −∞ , ∞ ) ( −∞ , ∞ )  .

 

5 )   Any constant solution  y ( x )  = c of the following equation is called -------- solution .

𝑑𝑦 / 𝑑𝑥  =  𝑓 ( 𝑦 )  .

A )   Stationary ,

B )   Equilibrium ,

C )   Critical ,

D )   None ,

Correct answers : -   B )   Equilibrium ,

 

6 )   For the following DE , any number of the family of curves  f ( x , y )  =  c ,   is called  . . . . . . . ,

𝑑𝑦 / 𝑑𝑥  =  𝑓 ( 𝑥 , 𝑦 )  .

A )  Distructive ,

B )  Dispersive ,

C )  Monochromatic ,

D )  Isocline ,

Correct answers : -   D )  Isocline ,

 

7 )  The period of simple harmonic motion weighing 8 pounds attached to a spring whose constant is 6.25 lb / ft  is ---------  second .

A )  𝑇 = 2𝜋 / 5  ,

B )  𝑇 = 𝜋 / 5  ,

C )  𝑇 = 3𝜋 / 5  ,

D )  𝑇 = 𝜋 / 4  ,

Correct answers : -   A )  𝑇 = 2𝜋 / 5  ,

 

8 )  Which of the following is a solution of the following equation ?

𝑦″ 𝑦 = 𝑦′^2  .

A )  𝑦 = 𝑒^(𝑐2𝑥) ,

B )  𝑦 = 𝑒^(𝑐2𝑥^2)  ,

C )  𝑦 = 𝑐4𝑒^(𝑐2𝑥)  ,

D )  None of the above ,

Correct answers : -   C )  𝑦 = 𝑐4𝑒^(𝑐2𝑥)  ,

 

9 )  The following equation represents a spring/mass system .  What is the amplitude of vibrations in meter , when a body is released initially from a point 1m above the equilibrium position with a downward velocity of 3 m / s ?

𝑥″ + 16𝑥 = 0  .

A )  5 ,

B )  3 / 4 ,

C )  1 / 2 ,

D )  5 / 4 ,

Correct answers : -   D )  5 / 4 ,

 

10 )  Which of the following is a solution of the following D.E. ?

( 2𝑦 – 2 ) 𝑦′ = 2𝑥 – 1  .

A )  𝑦^2 − 2𝑦 = 𝑥^2 – 𝑥 + 𝑐  ,

B )  𝑦^2 + 2𝑦 = 𝑥2 + 𝑥 + 𝑐  ,

C )  𝑦^2 − 2𝑦 = 𝑥2 + 𝑐  ,

D )  𝑦^2 = 𝑥^2 – 𝑥 + 𝑐  ,

Correct answers : -   A )  𝑦^2 − 2𝑦 = 𝑥^2 – 𝑥 + 𝑐  ,

 

11 )  What is the Laplace of the following function ?

𝑓 ( 𝑡 )  =  −1  ,  𝑖𝑓 0 ≤ 𝑡 < 1 𝑎𝑛𝑑  𝑓 ( 𝑡 )  =  1   ,  𝑖𝑓 𝑡 ≥ 1  .

A )  ( 1 / 𝑠 ) ( 2𝑒^(−𝑠) – 1 )  ,

B )  ( 2𝑒^(−𝑠) – 1 )  ,

C )  ( 1 / 𝑠 ) ( 2𝑒^(−𝑠) + 1 )  ,

D )  ( 2𝑒^(−𝑠) + 1 )  ,

Correct answers : -   A )  ( 1 / 𝑠) ( 2𝑒^(−𝑠) – 1 )  ,

 

12 )  What is the radius of convergence of the following power series ?

∞∑𝑛=0    𝑎𝑛 ( 𝑥 + 1 ) ^ 𝑛     .

A )   0  ,

B )   √8  ,

C )   4  ,

D )  ∞ ,

Correct answers : -   B )   √8  ,

 

13 )  What is the inverse Laplace of the following function?

𝐿 ^ (−1)   { 1 / 𝑠 ^ 5 } = ? 

A )  𝑡 ^ 4  ,

B )  ( 𝑡 ^ 5 ) / 120  ,

C )  ( 𝑡 ^ 2 ) / 2  ,

D )  ( 𝑡 ^ 4 ) / 24 ,

Correct answers : -   D )  ( 𝑡 ^ 4 ) / 24 ,

 

14 )  Which of the following are the singular point ( s )  of the following equation ?

(𝑥^2) 𝑦″ + 𝑥 𝑦′ + 2𝑦  =  0  ,

A )  ±1 ,

B )  1 ,

C )  0 ,

D )  ± 𝑖  ,

Correct answers : -   C )  0 ,

 

15 )   For what value of M ( x , y ) , the following equation becomes exact ?

𝑀 ( 𝑥 , 𝑦 ) 𝑑𝑥 + ( 𝑥 𝑒 ^ ( 𝑥 𝑦 ) + 2 𝑥 𝑦 + 1 𝑥 ) 𝑑𝑦 = 0  ,

A )  2 𝑥 𝑦 + 𝑒 ^ ( 𝑥 𝑦 )  ,

B )  2 𝑦 + 𝑦 𝑒 ^ ( 𝑥 𝑦 )  ,

C )  2 𝑥 𝑦 − 𝑦 𝑒 ^ ( 𝑥 𝑦 ) ,

D )  2 𝑥 𝑦 + 𝑦 𝑒 ^ ( 𝑥 𝑦 ) ,

Correct answers : -    D )  2 𝑥 𝑦 + 𝑦 𝑒 ^ ( 𝑥 𝑦 ) ,