Differential Equations and Vector Calculus

UNIT-I
𝑑2 𝑦 𝑑2 𝑦
1) The order of 𝑥 3 𝑑𝑥 2 + 2𝑥 2 𝑑𝑥 2 − 3𝑦 = 𝑥 is _______________ [ A ]
A) 2 B) 3 C) 1 D) None
3
𝑑2 𝑦 𝑑𝑦 2 2 𝑑2 𝑦
2) The degree of the differential equation [𝑑𝑥 2 + (𝑑𝑥 ) ] = 𝑎 𝑑𝑥 2 is [ B ]
A) 2 B) 3 C) 1 D) None
𝑑𝑦 𝑥+𝑦
3) The general solution of 𝑑𝑥 = 𝑒 is [ B ]
𝑥 𝑦
A) 𝑒 + 𝑒 = 𝐶 B) 𝑒 𝑥 + 𝑒 −𝑦 = 𝐶 C) 𝑒 −𝑥 𝑦
+𝑒 =𝐶 D) 𝑒 −𝑥
+ 𝑒 −𝑦 = 𝐶
4) What is the condition of exact differential equations [ C ]
M N M N M N M N
A)  B)  C)  D) 
x y x y y x y x
𝑦𝑑𝑥−𝑥𝑑𝑦
5) The general solution of = 0 is [ B ]
𝑦2
A) xy=c B) x=cy C) y=xc D) x=c𝑦 2
𝑥𝑑𝑥+𝑦𝑑𝑦
6) The general solution of 2 2 = 0 is [ D ]
𝑥 +𝑦
A) log(𝑥 + 𝑦)=c B) log(𝑥 − 𝑦)=c C) log(𝑥, 𝑦)=c D)None
( 2) ( 2)
7) The general solution of 1 + 𝑥 𝑑𝑦- 1 + 𝑦 𝑑𝑥=0 [ A ]
A)tan−1 𝑦 − tan−1 𝑥 = 𝑐 B) tan−1 𝑦 𝑥 = 𝑐 C)cos −1 (𝑥 + 𝑦) =c D)None
8) If the equation y f x, y dx  x g x, y dy  0 and Mx-Ny≠ 0 then the integrating factor is ______
[ A ]
1 1
A) 𝑀𝑥−𝑁𝑦 B) 𝑀𝑥+𝑁𝑦 C) Mx+Ny D)None
9) If the equation Mdx+Ndy=0 Homogeneous DE and Mx+Ny≠ 0 then the integrating factor is ______
[ B ]
1 1
A) 𝑀𝑥−𝑁𝑦 B) 𝑀𝑥+𝑁𝑦 C)Mx+NY D)None
𝜕𝑀 𝜕𝑁
−
𝜕𝑦 𝜕𝑥
10) If there exists a continuous single variable function f(x) such that = 𝑓(𝑥) then
𝑁
____________ is the integrating factor of Mdx+Ndy=0 [ A ]
A)𝑒 ∫ 𝑓(𝑥)𝑑𝑥 B)𝑒 ∫ 𝑓(−𝑥)𝑑𝑥 C) 𝑒 𝑓(𝑥) D) None
𝜕𝑁 𝜕𝑀
−
𝜕𝑥 𝜕𝑦
11) If there exists a continuous single variable function f(x) such that 𝑀 = g(y) then
____________ is the integrating factor of Mdx+Ndy=0 [ D ]
A)𝑒 ∫ 𝑔(𝑥)𝑑𝑥 B)𝑒 ∫ 𝑔(𝑦)𝑑𝑥 C) 𝑒 𝑓(𝑥) D) None
𝑑𝑦
12) The general solution of linear differential equation 𝑑𝑥 +P(x).y=Q(x) is [ A ]
A)𝑦𝑋(𝐼. 𝐹 ) = ∫ 𝑄 𝑋(𝐼. 𝐹 )𝑑𝑥 + 𝑐 B)𝑥𝑋(𝐼. 𝐹 ) = ∫ 𝑄 𝑋(𝐼. 𝐹 )𝑑𝑦 + 𝑐
C)xdy-ydx=c D)None
13) What is the form of Benrnoulli’s DE equation [ C ]
A)  Px y  Qx  B)  Px y  Qx  x n C)  Px y  Qx y n
dy dy dy
D) None
dx dx dx
dy 1
14) Find IF for  y  sin 2 x [ A ]
dx x
1 1
A) B) - C) −𝑥 D) x
x x
dy 1
15) Find IF for  y  x 2 log x [ B ]
dx x log x
A) loglog x  B) log x C) loglog x 2  D) None
dy 1
16) Find IF for  y  x2 y2 [ D ]
dx x
1 1
A) B) - C) −𝑥 D) x
x x
17) Define IF __________________
18) What is the condition for Non-Exact DE_______________
19) State Newton’s Law of cooling_________________
20) What is the equation in Law of natural decay__________

Unit - II
d3 y
1. The P.I of 3
 y  e x is ------ [ B ]
dx
x
a) xe  x /3 b) e  x c)  xe  x /3 d) None
3
2. C.F of ( D  1)( D  2) y  e3 x is
2
[ A ]
a) c1e x  (c2  c3 x)e 2 x b) c1e x  (c2  c3 x)e 2 x
b) c) c1e  x  (c2  c3 x 2 )e 2 x d) c1e x  (c2  c3 x)e 2 x
3. P. I of ( D 2  a 2 ) y  cos ax is [ C ]
x x
a) cos ax b) x cos ax c) sin ax d) x sin ax
2a 2a
4. P.I of ( D  1)2 y  x is [ D ]
b)  x  1 c)  x  2  d) x  2
2 2
a) x
1
5. sin x  [ A ]
D  D 1
2

1
a)  cos x b) sin x c) sin x d) sin x
4
d3 y
6. The general solution of  y  0 is [ B ]
dt 3
  3t 
t
3t
a) y  c1et  e 2  c2 cos  c3 sin 
 2 2 

t
 3t 3t 
b) b) y  c1e  e t 2
 c2 cos  c3 sin 
 2 2 
 3t 3t 
c) y  c1et  e  t  c2 cos  c3 sin 
 2 2 
 3t 3t 
d) y  c1et /3  e  t  c2 cos  c3 sin 
 2 2 
7. The P.I of y   3 y   2 y  12 is [ C ]
a) 5 b) -6 c) 6 d) 2
8. The C.F of y   2 y   y  xe sin x is
x
[ C ]
a) c1e 2 x b) ce  x c)  c1  c2 x  e x d)  c1  c2 x 2  e x

 
9. e x c1 cos 3 x  c2 sin 3 x  c3 e2 x is the general solution of [ B ]

d3 y d3 y d3 y d3 y d 2 y dy
a)  4y  0 b)  8y  0 c)  8y  0 d)  2   2y  0
dx3 dx3 dx3 dx3 dx 2 dx
10. The general solution of  D 2  1 y  0 is
2
[ A ]
a) y   c1  c2 x  cos x   c3  c4 x  sin x b) y  c1 cos x  c2 sin x
 c) y  c1e x  c2 e  x d) y   c1  c2 x  e x
d2 y
11. The P.I of  y  cosh 3x is [ C ]
dx 2
cosh 3 x sinh 3 x
a) cosh 3x b) sinh3x c) d)
10 10
12. P.I of  D  1 y  e x sin x is
2
[ C ]
a) e x cos x b) e  x cos x c) e x sin x d) e x sin x
1
13. If F ( D)  D 2  5, sin 2 x        [ B ]
F ( D)
a) cos 2x b) sin 2x c)  cos 2x d)  sin 2x
14. The P.I of  D 2  4 D  4  y  2 x is [ A ]
a) 0.3464 b) 0.5464 c) 0.7 d) 0.9
15. The general solution of  4 D  4 D  D  y  0 is
3 2
[ B ]
x x
a) y  c1e x   c2  c3 x  e 2 b) y  c1   c2  c3 x  e 2

y  c1e x   c2  c3 x  e x
c) d) y  c1e x   c2  c3 x  e x
16. P.I of F ( D) y  Q( x) where Q( x)  x v( x) , v ( x ) is a function of x alone is [ A ]
 1  1  1  1
a)  x  F ( D)  v( x) b)  x  F ( D)  v( x)
 F ( D)  F ( D)  F ( D)  F ( D)

 1  1  1  1
c)  x  d)  x 
F ( D)  F ( D) F ( D)  F ( D)
v( x) v( x)
 
17. If c1u  x   c2 v  x  is the complementary function of differential equation
d2 y dy
2
 P( x)  Q( x) y  R( x) then its particular integral according to the method of variation
dx dx
of parameters is [ A ]
a) A( x)u  x   B  x  v( x) b) xu  x   x v( x)
2

b) c) xu  x   x2 v( x) d) None of the above

d2 y dy
18. For the given differential equation 2
 P( x)  Q( x) y  R( x) , C.F = c1u  x   c2 v  x 
dx dx
& P.I = A( x)u  x   B  x  v( x) then which of the following is correct [ B ]
vR uR
a) A( x)    dx; B( x)  
dx
dv du dv du
u v u v
dx dx dx dx
vR uR
b) A( x)    dx; B( x)   dx
dv du dv du
u v u v
dx dx dx dx
vR uR
c) A( x)   dx; B( x)    dx
dv du dv du
u v u v
dx dx dx dx
uR vR
d) A( x)    dx; B( x)   dx
dv du dv du
u v u v
dx dx dx dx
 d2 y
19. For the given differential equation 2
 a 2 y  cos ec ax , C.F = c1 cos ax  c2 sin ax &
dx
P.I = A( x)cos ax  B  x  sin ax then the value of A( x ) is [ A ]
x x
a) b) c) 0 d)  x
a a
d2 y
20. For the given differential equation 2
 a 2 y  sec ax , C.F = c1 cos ax  c2 sin ax &
dx
P.I = A( x)cos ax  B  x  sin ax then the value of B ( x ) is [ B ]
x x
a) b) c) 0 d)  x
a a