Unit-I

dy 3
1. Solve + y tan x=cos x .
dx
2. Solve y ( xy sin xy +cos xy ) dx+ x ( xy sin xy −cos xy ) dy=0 .
dy
3. Solve x + y=log x .
dx
4. Define an Integrating Factor.
5. Solve ( y 2−2 xy ) dx + ( 2 xy−x 2 ) dy=0 .
6. State Newton’s law of cooling.
dy 2 3
7. Solve x ( x−1 ) − y =x ( x−1 ) .
dx

8. Solve
( y)
( 1+e x / y ) dx +e x / y 1− x dy=0
.
tan−1 y
dx 1 e
+ x=
9. Solve dy 1+ y
2
1+ y 2 .
10. Solve x 2 ydx −( x 2+ y 2) dy =0.
11. A body is originally at 80 o C and cools down to 60 o C in 20 minutes. If the
temperature of the air is 40 o C , find the temperature of the body after 40 minutes.
12. If the air is maintained at 15o C and the temperature of the body drops from 70o C to
o
40 C in 10 minutes. What will be its temperature after 30 minutes.
13. The number N of bacteria in culture grew at a rate proportional to N. The value of N
1
was initially 100 and increased to 332 in one hour. What was the value of N after 1
2
hours.
14. A bacterial culture growing exponentially increases from 200 to 500 grams in the
period from 6 a.m. to 9 a.m. How many grams will be present at noon.
15. Find the charge in RC circuit if R=20 ohms, C=0.01 farad and E ( t )=20 sin 2 t with
q ( 0 )=0.

Unit-II
1. Solve ( D3−1 ) y =0.
2. Solve y ″ −3 y ' + 2 y =0.
3. Solve y ' ' +6 y ' +9 y =0 , y ( 0 )=−4∧ y' ( 0 )=14 .
4. Solve ( D −1 ) y=0 .
2

1 2x
5. Find e .
D−2
6. Find particular integral of ( D2 +5 D+6 ) y =e x .
7. Solve ( D2 + 4 D+3 ) y=e2 x .
8. Solve ( D2−3 D+2 ) y =cos 3 x .
9. Solve D2 ( D2+ 4 ) y=320 ( x 3+ 2 x 2 +e x ).
10. Solve ( D2−1 ) y =x e x sin x .
11. Apply the method of variation of parameters to solve ( D2 +a 2 ) y =tan a x .
12. Apply the method of variation of parameters to solve ( D2 + 4 ) y=tan 2 x .
13. Apply the method of variation of parameters to solve ( D2 +a 2 ) y =sec ax .
14. Apply the method of variation of parameters to solve ( D2 +1 ) y=co sec x .
15. Apply the method of variation of parameters: ( D2 +3 D+2 ) y=e x + x 2.

Unit-III
1. Form the partial differential equation z=ax+by + a2 + b2 by eliminating the
arbitrary constants a and b .
2. Form the partial differential equation by eliminating the function f from the relation
f ( x 2 + y 2 + z 2 , xyz ) =0.
3. Solve p+q=1.
4. Solve p x 2+ q y 2=z 2.
5. Solve x p− yq= y 2−x 2.
6. Find the differential equation of all spheres whose center lie on the z -axis and given
by equation x 2+ y 2+ ( z −a )2=b2 , a∧b being constants.
7. Form the partial differential equation by eliminating the constants ‘a ’ and ‘b ’ from

z=a log [ b ( y−1 )

1−x ] .

8. Form the partial differential equation by eliminating the arbitrary function from
z=( x + y ) ϕ ( x 2− y 2) .
9. Form the partial differential equation by eliminating the arbitrary functions from
z=f ( x + at ) + g ( x−at ) .

10. Form the PDE from the relation z=f ( x +¿ )+ g ( x−¿ ).

11. Solve x 2 ( y −z ) p+ y 2 ( z−x ) q=z 2 ( x− y ).

12. Solve ( x 2− yz ) p+ ( y 2−zx ) q=z 2−xy .
2 2
13. Solve z ( x − y )=x p− y q .

14. Solve ( D
2
+ 4 DD
'
−5 D
'2
) z=sin ( 2 x +3 y ) .
2 2
∂ z ∂ z
15. Solve 2 − =sin x cos 2 y.
∂ x ∂ x∂ y

Unit-IV
1. Find grad ϕ where ϕ ( x , y , z ) =log ( x + y 2+ z 2 ).
2

2. Prove that F= yz i+zx j+xy k is irrotational.

2 2 2 3
3. If F=x zi−2 y z j+xy z k then find div F .
4. If F=( x +3 y ) i+ ( y−2 z ) j+ ( x+ pz ) k is solenoidal, find p.
5. Find the directional derivative of ϕ¿ x y + y z in the direction of vector i+2 j+2 k at
2 2

the point (2, -1, 1). In what direction it will be maximum. Find its maximum value.
6. Find the directional derivative of f =x 2− y 2 +2 z2 at the point P= (1 , 2 ,3 )in the
direction of the line PQ where Q= (5 , 0 , 4 ) . In what direction it will be maximum.
Find its maximum value.
2 '
7. Show that ∇ [ f ( r ) ] =f ( r ) + f ( r ) where r =|r|.
2 ''
r
2
8. Prove that curl curl F=grad ( div F )−∇ F .
9. If r = xi+ y j+z k then find ¿ r ∧curl r .
10. Prove that div ( grad r ) =n ( n+1 ) r where r = xi+ y j+z k .
n n−2

11. Find the directional derivative of f =xy + yz + zx in the direction of vector i+2 j+2 k
at the point (1, 2, 0).
12. In what direction from (3, 1, -2), direction derivative of f =x 2 y 2 z 4 is maximum. Find
the maximum value.
13. If F=x ( y + z ) i+ y ( z + x ) j+ z ( x + y ) k then find ¿ F and curl F .
14. Find ∇ 2 r n.
15. Find the angle between the surfaces x 2+ y 2+ z 2=9 and z=x 2 + y 2−3 at the point
( 2 ,−1 , 2 ).

Unit-V
1. Define Line integral.
2. Define Surface integral.
3. Define Volume integral.
4. State Green’s theorem.
5. State Stok’s theorem.
6. State Gauss’s Divergence theorem.

∫ ( y 2 dx−2 x 2 dy ) 2
7. Evaluate C along the parabola y=x from (0, 0) to (2, 4).
❑

8. Evaluate ∫ ( x dy− y dx ) around the circle c : x 2+ y 2 =1.

c
❑

9. the|line integral∫ [ ( x + xy ) dx + ( x + y ) dy ] where C is the square formed by the lines

2 2 2

x=± 1 , y =±1.
❑

10. the|line integral∫ [ ( x + y ) dx−2 xydy ] whereC is the rectangle bounded by the lines
2 2

x=± a , y=0 , y =b.

11. Verify stokes theorem for a vector field defined by F=( x 2− y 2 ) i+ 2 xy j in a
rectangular region in the XOY plane bounded by the lines x=0 , x=a , y =0 , y=b .
 ∮ [ ( 3 x 2−8 y 2) dx +( 4 y−6 xy ) dy ]
12. Verify Green’s theorem in plane for C where C is the
region bounded by y= √ x and y=x 2 .
∮ [ ( 3 x 2−8 y 2) dx +( 4 y−6 xy ) dy ]
13. Verify Green’s theorem in plane for C where C is the
region bounded by x=0 , y=0 and x + y=1.
❑

14. by|Gree n s theorem∮ [ ( y−sin x ) dx +cos x dy ] whereC isthe triangle enclosed by the
'

π
lines y=0 , x= , πy=2 x .
2
15. Verify Gauss’s divergence theorem for F=( x 2− yz ) i+ ( y 2−zx ) j+ ( z 2−xy ) k taken over
the rectangular parallelepiped 0 ≤ x ≤ a , 0 ≤ y ≤ b , 0 ≤ z ≤ c.