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 ) ,
