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 ) ,
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