1. AMET University
Kanathur, Chennai.
Subject: Mathematics – II
Assignment
Unit I: Ordinary Differential equation
Solve the following differential equation:
dy
1. x4
dx
+ x 3 y = − sec ( xy ) 2. (1 + y 2 ) dx = ( tan −1 y − x ) dy
2 dy
3. (1 − x ) + 2 xy = x 1 − x 2 given that x=0 when y=0.
dx
Bernoulli’s equation:
dy tan y dy
4. − = (1 + x)e x sec y 5. xy (1 + xy 2 ) = 1
dx 1 + x dx
2
dy sin x cos x
6. solve − y tan x =
dx y2
Exact equation:
7. (x 2
y 2 + xy + 1) ydx + ( x 2 y 2 − xy + 1) xdy = 0 8. ( xy 2 3
)
− e1/ x dx − x 2 ydy = 0
x x
x
9.. (1 + e )dx + (1 − )e y dy = 0
y
y
Equations first order and higher degree:
10. x − yp = ap 2 11. ( px − y )( py + x ) = a 2 p
Application of Differential equation:
12. r = a (1 − cos θ )
Find the orthogonal trajectory of the cardioids
13. A moving body is opposed by a force per unit mass of value cx and resistance per
2
unit of masses of value bv where x and v the displacement and velocity of the
particle are at that instant. Find the velocity of the particle in terms of x, if it starts
from rest.
14. Show that the differential equation for the current i in an electrical circuit
containing an inductance L and a resistance R in series and acted on by an
di
electromotive force E sin ω t satisfies the equation L + Ri = E sin ω t . Find the
dt
value of the current at any time t, if initially there is no current in the circuit.
Higher order equations with constant coefficient:
d2y dy
Solve the following 15.
dx 2
− 2 + y = xe x sin x
dx
16. (D 2
+ 4 ) y = e x sin 2 x
17. ( D + 4D + 3) y = e sin x + x
2 −x
Method of Variation of parameter:
d2y
Solve the following 18. + y = cos ec x 19. y ''− 2 y '+ 2 y = e x tan x
dx 2
20. solve ( D 2 + a 2 ) y = sec ax
1

2. Unit II Partial Differential equations
Form the PDE by eliminating the arbitrary function:
1. z = f1 ( y + 2 x) + f 2 ( y − 3 x) 2. f ( x 2 + y 2 , z − xy ) = 0 3. z = f (x2 + y2 + z2 )
4. xyz = φ ( x 2 + y 2 − z 2 ) 5. z = φ ( x)φ ( y )
Form the PDE by eliminating the arbitrary constants from:
6. z = ax + by + a 2 + b 2 7. z = axe y + a 2 e 2 y + b 8. log(az − 1) = x + ay + b
Solution of PDE by direct integration:
∂2 z ∂z ∂2 z
9. Solve = z , gives that when y = 0, z = e x and = e− x 10. Solve = xy
∂y 2
∂y ∂x 2
Lagrange’s equation:
Solve the following
11. (x 2
− yz ) p + ( y 2 − zx ) q = z 2 − xy 12. xp + yq = 3 z
13. (x 2
− y 2 − z 2 ) p + 2 xyq = 2 xz
Non-linear equations of the first order
14. z = p2 + q2 15. z 2 ( p2 + q2 ) = x2 + y2 16. p + q = x + y.
Four types - f ( p, q) = 0, f ( z , p, q) = 0, f ( x, p ) = f ( y, q ) and z = xp + yq + f ( p, q)
17. solve p 2 + q 2 = 1 solve p + q = 1 18. solve z = px + qy + p 2 q 2
19. solve z = p 2 + q 2 for complete and Singular solutions.
20. solve p 2 + q 2 = x + y
Unit III Algebra of Matrices
System of linear equations:
1. Find the values of λ , for which the following system of equations is consistent and
has non trivial solution, solve equations for all such values of λ , ( λ -1) x+(3 λ +1)
y +2 λ z =0, ( λ -1) x+(4 λ -2) y+( λ +3) z= 0, 2x+(3 λ +1) y+3( λ -1) z=0.
Test the consistency of the following equations and solve them if possible:
2. 3x+3y+2z=1, x+2y=4, 10y+3z=-2, 2x-3y-z=5.
3. 4x-5y+z=2, 3x+y-2z=9, x+4y+z=5.
4. Determine the values of a and b for which the system
 3 −2 1  x   b  (i) has a unique solution
     (ii) has no solution and
 5 −8 9  y  =  3  (iii) has infinitely many solutions.
 2 1 a  z   −1
    
5. Find the characteristic equation of the matrix
3 1 1 Verify Cayley Hamilton theorem for this matrix.
 
A =  −1 5 −1 Hence find A− 1
 1 −1 3 
 
Eigen values and vectors:
Find the Eigen values and the corresponding Eigen vectors for the following matrices:
 1 −6 −4   2 −2 2   2 −1 1   3 −1 1 
       
6. A =  0 4 2  7. A =  1 1 1  8. A =  −1 2 −1 9. A =  −1 5 −1
 0 −6 −3   1 3 −1  1 −1 2   1 −1 3 
   2    

3. Unit IV Differentiation of vectors
1. If P = 5t 2 I + t 3 J − tK and Q = 2 I sin t − J cos t + 5tK find
d d
( i ) ( P.Q ) ; ( ii ) ( P × Q )
dt dt
2. Find the curvature and torsion of the curve x = a cos t , y = a sin t , z = bt
Velocity and acceleration:
3. A particle moves along the curve R = ( t 3 − 4t ) I + ( t 2 + 4t ) J + ( 8t 2 − 3t 3 ) K where t
denotes time. Find the magnitudes of acceleration along the tangent and normal at
time t=2.
4. The velocity of a boat relative to water is represented by 3I+4J and that of
water relative to earth is I-3J. What is the velocity of the boat relative to the earth
if I and J represent one km. an hour east and north respectively.
−t
5. A particle moves along a curve x = e , y = 2 cos 3t , z = 2 sin 3t , where t is the time
variable. Determine its velocity and acceleration vectors and also the magnitudes
of velocity and acceleration at t =0.
Del – curl:
Prove that the following:
6. ∇ ⋅ ( F × G ) = G ⋅ (∇ × F ) − F ⋅ (∇ × G ) 7. div curl F = ∇ ⋅∇ × F = 0
8. grad (V1 ⋅V2 ) = V1 + V2 9. curl ( A ⋅ R ) R  = A × R
 
2
∇ 2 f ( r ) = f '' ( r ) +f ' ( r ) 11. Show that ∇ ( r ) = n ( n + 1) r
2 n n−2
10.
r
11. If φ ( x, y, z ) = x y + y 2 x + z 2 find ∇φ at the point (1,1,1).
2
r r
12. If r = xi + y j + zk and r= r provethat (i ) ∇r =
r r ( )
(ii ) ∇ 1 = − 3
r
r r
(iii ) ∇r n = nr n − 2 r (iv)∇ ( log r ) = 2 (v) ∇ ( f (r ) ) = f ' (r ) = f ' (r )∇r.
r r
3
13. If F = xz i − 2 xyz j + xzk find divF and curl F at (1, 2,0).
14. showthat the vector 3 x 2 yi − 4 xy 2 j + 2 xyzk is solenoidal.
15. Find the value of ' a ' such that F = ( axy − z 2 ) i + ( x 2 + 2 y ) j + ( y 2 − axz )k is irrotational.
16. Showthat F = ( y 2 − z 2 + 3 yz − 2 x ) i + (3 xz + 2 xy ) j + (3 xy − 2 xz + 2 z )k is irrational and
solenoidal.
r 2
17. Pr ove that div   = .
r r
18. If r = xi + y j + zk show that ∇ × (r 2 r ) = 0
19. provethat ∇ 2 ( r n ) = n(n + 1) r n − 2 .
20. provethat ∇ × ( ∇r n ) = 0. ]
3

4. Unit V
Line integral – Work
1. Find the work done in moving particle in the force field F = 3x 2 I + ( 2 xz − y ) J + zK ,
along (a) the straight line from ( 0, 0, 0 ) to ( 2,1,3) . (b) the curve defined by
x 2 = 4 y, 3 x3 = 8 z from x=0 to x=2.
2. If F = 2 y i − z j + x k , evaluate ∫ C
F × dR along the curve
π
x = cos t , y = sin t , z = 2 cos t from t = 0 to t = .
2
Surface integrals:
3. Evaluate ∫ F ⋅ dS where F = xI + ( z 2 − zx ) J − xyK and S is the triangular surfaces
S
with vertices ( 2, 0, 0 ) , ( 0, 2, 0 ) and ( 0, 0, 4 ) .
4. If velocity vector is F=yI+2J+xzK m/sec., show that the flux of water through the
parabolic cylinder y = x 2 , 0 ≤ x ≤ 3, 0 ≤ z ≤ 2 is 69 m3/sec.
Green’s theorem:
∫ ( xy + y ) dx + x dy 
2 2
5. Verify Green’s theorem for where C is bounded by
C 
y = x and y = x 2 .
6. Using Green’s theorem evaluate  ∫
 x 2 ydx + x 2 dy  where C is the boundary
 C
described counter clockwise of the triangle with vertices ( 0, 0 ) , (1, 0 ) , (1,1) .
∫ ( x − 2 xy ) dx + ( x 2 y + 3) dy  around the boundary C
2
7. Verify Green’s theorem for
C 
of the region y 2 = 8 x and x = 2 .
8. Evaluate ∫∫ S
F ⋅ dS , where F = xyi − x 2 j + ( x + z ) k and S is region of the plane
2x+2y+2z=6 in first octant.
∫ ( xy − x ) dx + x
2 2
9. Compute y dy over the triangle bounded by the line a
C
y = 0, x = 1, y = x and verify Green’s theorem.
Stoke’s theorem:
10. Verify Stoke’s theorem for F = ( x 2 + y 2 ) i − 2 xy j taken around the rectangle
bounded by the lines x = ± a, y = 0, y = b.
11. Using Stoke’s theorem evaluate ∫ ( x + y ) dx + ( 2 x − z ) dy + ( y + z ) dz 
C  where C is
the boundary of the triangle with vertices ( 2, 0, 0 ) , ( 0,3, 0 ) , ( 0, 0,6 ) .
12. Apply Stoke’s theorem to calculate ∫C
4 ydx + 2 zdy + 6 ydz where C is the curve of
the intersection of x 2 + y 2 + z 2 = 6 z and z = x + 3 .
13. Verify Stoke’s theorem for F = ( y − z + 2 ) i + ( yz + 4 ) j − xz k over the surface of a
cube x=0, y=0, z=0, x=2, y=2, z=2 above the XOY plane (open the bottom).
14. Apply Stoke’s theorem to find the value of ∫ ( ydx + zdy + xdz )
C
where C is the curve
of intersection of x 2 + y 2 + z 2 = a 2 and x + z = a .
4

5. Gauss Divergence theorem:
15. Verify Divergence theorem, given that F = 4 xz i − y 2 j + yz k and S is the surface of
the cube bounded by the planes x=0, x=1, y=0, y=1, z=0, z=1.
∫∫ ( y z i + z x 2 j + x 2 y 2 k ) ds where S is the
2 2 2
16. Use divergence theorem to evaluate
upper part of the sphere x 2 + y 2 + z 2 = 9 above X-Y plane.
17. Verify divergence theorem for F = ( x 2 − yz ) i + ( y 2 − zx ) j + ( z 2 − xy ) k taken over
the rectangle parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.
∫ ( 2 xy i + yz j + xz k ) ds over the surface of the region bounded by x=0,
2
18. Evaluate
S
y=0, y=3, z=0 and x+2z=6.
19. Evaluate ∫ ( yz i + zx j + xy k ) ds
S
where S is the surface of the sphere
x 2 + y 2 + z = a 2 in the first octant.
2
20. Verify Gauss divergence therorem for the function F = 4 xi − 2 y 2 j + z 2 k taken over
the region boundeb by x 2 + y 2 = 4, z = 0, z = 3.
Prepared by, P.Suresh kumar
5