SNS COLLEGE OF ENGINEERING, CBE – 107

QUSTION BANK

MATHEMATICS – II

UNIT-1 (ORDINARY DIFFERENTIAL EQUATIONS)

PART-A

1) Define linear differential equations

2) Solve d2 ydx2 −¿.6

dydx +13y = 0

3) Solve (D2 + 1) y =0 given y(0) =0 , y’(0)=14) Solve (4D2-4D +1)y = 45) Solve (D2_1)y = x6) Solve the equation x2 y11-xy1+y= 07) Solve (D+2)2y = e−2xsinx8) Find the particular integral of (D2+4)y = cos2x9) Find the particular integral of (D2+1)y = sinx10) Find the particular integral of (D2+1)y = xex

11) Find the particular integral of (D2_4)y = cosh2x12) Find the particular integral of (D-1)2y = exsinx13) Find the particular integral of (D2-2D+5)y = ex cos2x 14) Find the particular integral of (D2-2D+1)y = ex ( 3x2-2) 15) Find the particular integral of (D2-4D+4)y = 2x 16) Solve (D2+2D +1) y =π17) Solve (D2-3D -4) y = e 3x +e-x

18) (D2+1) y = sin2x19) Transform the equation x2 y11+xy1= x into a linear differential equation with constant Co-efficient.

20) Transform the equation (2x+3)2 d2 ydx2 - 2(2x+3)

dydx - 12y =6x into a linear differential

Equation with constant co-efficients21) Find the Wronskian of y1, y2 of y11 -2y 1 +y =ex logx

Part-B

1) Find the particular integral of (D2+4)y = x2 cos2x2) Solve (D2+3D+2)y = sin3x cos2x3) Solve (D2+5D+6)y = e –7x sinh3x4) Solve (D3 - 3D2 + 4D-2)y = sinh2x

1

5) (D2+9)y = 11 cos3x6) Solve (D2-6D+13)y =8 e 3x sin4x7) Solve (D2+4D+4)y = e –2x/ x2 8) Solve (D2-2D+1)y = e x xcosx9) Solve the equation (D2+a2)y = secax by the method of variation of parameters

10) Solved2 ydx2 +¿ 4y= 4tan2x by the method of variation of parameters

11) Solved2 ydx2 +¿ y= cosecx by the method of variation of parameters

12) Solve (D2+1)y = xsinx by the method of variation of parameters13) Solve (D2-2D+1)y = e x logx by the method of variation of parameters

14) Solve(1+x2)2d2 ydx2 +(1+x)

dydx

+ y = 2sin(log(1+x))

15) Solve (x2 d2 ydx2 -2x

dydx -4)y = x2+2logx

16) Solve(x2D2+ xD+1)y= logxsin(logx)17) Solve(x2D2+ xD+4)y=cos(logx)+ xsin(logx)

18) Solve (D2+1XD)y =

12logxx2

19) Solve(x2D2+ 3D+1)y= sin(logx)/x2

20) Solve((3x+2)2d2 ydx2 +3(3x+2)

dydx -36)y =3x2+4x+1

21) Solve(x+1)3d2 ydx2 +3(x+1)2dy

dx +(x+1)y =6log(x+1)

22) Solve the system of equations dxdt +y =et ;x−

dydt = t

23) Solve the system of equations dxdt +y =sint ;

dydt +x=cost given that x=2,y=0 when t=0

24) Solve the system of equations dxdt +2x+3y =2e2t ;

dydt +3x+2y =0

25) Solve the system of equations dxdt

− y =t ;dydt

+¿x =t2

26) Solve the system of equationsdxdt

−dydt +2y=cos2t ;

dxdt

+ dydt -2x =sin2t

27) Solve the simultaneous differential equations dxdt +2y=sin2t ;

dydt -2x =cos2t

2

UNIT II – (VECTOR CALCULUS)

Part – A

1. Define vector Differential operator(∇)2. Define gradient of the scalar function φ.3. If f and g are two scalar point function then ∇ ¿fg) = f ∇ g + g∇f.4. If φ = log(x2+ y2+z2 ¿ find ∇ φ.

5. Prove that ∇ f (r )= f ' (r )rr⃗ ,r⃗ = x i⃗ + y j⃗ +z k⃗ .

6. Find the directional derivative of φ=xy+ yz+zx in the direction vector i⃗+2 j⃗+2 k⃗ at (1, 2 ,0)

7. Find the directional derivative of φ=3 x2+2 y−3 z at (1 , 1 ,1) in the direction of 2i⃗+2 j⃗−k⃗ .

8. Find the unit vector normal to the surface x2− y2+z=2 at the point (1 , -1 ,2)9. Find the unit vector normal to the surface x2 y+2 x z2=8 at the point(1 , 0, 2)10. Prove that ∇× ¿ )¿∇× F⃗ ±∇×G⃗.11. Prove that ∇ ∙ ( F⃗× G⃗ )=G⃗ ∙ (∇×F⃗ ) – F⃗ ∙ (∇×G⃗ ) .12. Find ∇ ∙ F⃗ and ∇× F⃗ of the vector point function F⃗=x z3 i⃗−2x2 y z j⃗+2 y z4 k⃗ at the point

(1 ,-1 ,1).13. Prove that curl ( grad φ) = 0.14. Prove that div (grad φ) = ∇2φ .15. Show that the vector F⃗=3 y4 z2 i⃗+4 x3 z2 j⃗−3 x2 y2 k⃗ is solenoidal and

2 xy i⃗+(x2+2 yz ) j⃗+( y2+1) k⃗ is irrotational.16. Find the value of a , b , c so that the vector

F⃗=( x+2 y+az ) i⃗+(bx−3 y−z ) j⃗+(4 x+cy+2 z )k⃗ is irrotational.

17. If F⃗=x2 i⃗+xy j⃗ , evaluate ∫ F⃗ ∙ d⃗r from (0 ,0) to (1 ,1) along the line y = x.18. Find the value of ‘a’ given two vectors 2 i⃗−3 j⃗+5 k⃗ and 3 i⃗+a j⃗−2 k⃗ are perpendicular.19. If r⃗=x i⃗+ y j⃗+z k⃗ . S is the upper half surface of the sphere x2+ y2+z2=a2,then find

∬S

r⃗ ∙n̂ ds

3

20. If v is the volume of the region enclosed by the cube 0 < x ,y ,z <1 and F⃗=x2 i⃗+ y2 j⃗+z2 k⃗

, then ∭V

∇ ∙ F⃗ dV is

Part – B

1. If r⃗=x i⃗+ y j⃗+z k⃗ prove that (i) ∇ r= r⃗r , (ii) ∇ rn=nrn−2 r⃗ ,where r=¿ r⃗∨¿¿.

2. Find the angle of intersection at the point (2 ,-1,2) of the surfaces x2+ y2+z2=9 and z=x2+ y2−z−3.

3. Find ‘a’ and ‘b’ such that the surfaces a x2−byz=(a+2 ) x and 4 x2 y+z3=4 cut orthogonally at (1 ,-1,2).

4. If ∇ φ=2xyz i⃗+x2 z j⃗+x2 y k⃗ , find the scalar potential φ.

5. Evaluate ∫C

❑

F⃗ ∙ d⃗r where F⃗=3 x2 i⃗+(2xz− y ) j⃗+ z k⃗ and C is the straight line from

A(0 ,0 ,0) to B(2 , 1, 3).

6. Given the vector field F⃗=xz i⃗+ yz j⃗−z2 k⃗ , evaluate ∫C

❑

F⃗ ∙ d⃗r from the point (0,0,0) to

(1,1,1) where C is the curve (i) x = t , y = t 2, z = t 3, (ii) the straight path from (0,0,0) to (1,1,1).

7. Find the total work done in moving a particle in a force field given by F⃗=(2 x− y+z ) i⃗+ ( x+ y−z ) j⃗+(3 x−2 y−5 z ) k⃗ along a circle C in the XY plane x2+ y2=9 , z=0.

8. Find the work done by the force F⃗=(2 xy+z3) i⃗+x2 j⃗+3x z2 k⃗ when it moves a particle from (1,-2,1) to (3,1,4) along any path.

9. Evaluate ∬S

F⃗⋅n̂ ds where F⃗= yz i⃗+zx j⃗+xy k⃗ . and S is that part of the surface of the

sphere x2+y2+z2 = 1 which lies in the first octant.

10. Evaluate ∬S

F⃗⋅n̂ ds where F⃗=18 z i⃗−12 j⃗+3 y k⃗ as S is the part of the plane 2x + 3y +

6z = 12 which is in the first octant.

11. Evaluate ∬SF⃗⋅n̂ ds

where F⃗=( x+ y2 )i⃗−2x j⃗+2 yz k⃗ where S is the region bounded by 2x + y + 2z = 6 in the first octant.

4

12. If F⃗=(2 x2−3 z )i⃗−2 xy j⃗−4 x k⃗ , then evaluate (i) ∭V

∇×F⃗ dV (ii)

∭V

∇⋅F⃗ dV,where V

is the region bounded by x = 0 , y = 0 , z = 0 and 2x + 2y + z = 4.13. Verify the Gauss divergence theorem for F⃗=4 xz i⃗− y2 j⃗+ yz k⃗ over the cube bounded by

x = 0 , x = 1, y = 0, y = 1, z = 0, z = 1.14. Verify the Divergence theorem for F⃗=(x2− yz ) i⃗+( y¿¿2−zx) j⃗+(z2−xy) k⃗ ¿ taken over

the rectangular parallelepiped 0 ≤ x ≤ a ,0 ≤ y ≤ b , 0 ≤ z ≤ c .

15. Evaluate ∬SF⃗⋅n̂ ds

where F⃗=4 xz i⃗− y2 j⃗+ yz k⃗ . and S is the surface of the cube bounded by x = 0 ,x = 1, y = 0, y = 1, z = 0, z = 1.

16. Use divergence theorem to evaluate F⃗=4 x i⃗−2 y2 j⃗+ z2 k⃗ and S is the surface bounding the region x2 + y2 = 4 z = 0 and z = 3.

17. Verify Green’s theorem in a plane for the integral ∫C

{( x−2 y )dx+ xdy },taken around the

circle x2 + y2 = 1.

18. Verify Green’s theorem for ∫C

{( x2− y2)dx+2 xydy } , where C is the boundary of the

rectangle in the XOY – plane bounded by the lines x = 0,x = a, y = 0 and y = a.

19. Verify Green’s theorem for ∫C

{( xy+ y2 )dx+x2dy }, where C is the closed curve of the

region bounded by y = x and y = x2 .

20. Using Green’s theorem, evaluate ∫C

{( y−sin x )dx+cos xdy },where C is the triangle

bounded by y=0 , x= π2, y=2 x

π.

21. By applying Green’s theorem prove that the area bounded by a simple closed curve C is = 12∫C

(xdy− ydx ) and hence find the area of the ellipse.

22. Verify Stoke’s theorem for a vector defined by F⃗=(x2− y2) i⃗+2 xy j⃗ in the rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.

23. Verify Stoke’s theorem for a vector defined by F⃗= y i⃗+z j⃗+x k⃗ ,where S is the upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.

24. Evaluate the integral ∫C

{( x+ y )dx+(2x−z )dy+( y+z )dz } where C is the boundary of

the triangle with vertices (2,0,0), (0,3,0) and (0,0,6) using Stoke’s theorem.

25. Evaluate ∫C

( xydx+ xy2 dy ) by the Stoke’s theorem where C is the square in the XY plane

with vertices (1,0), (-1,0), (0,1) and (0,-1).

5

26. Prove that ∫ r⃗×d⃗r=2∬

Sn̂ ds

where S is the surface enclosing a circuit C.

UNIT 3 - (ANALYTIC FUNCTIONS – COMPLEX VARIABLES)

Part A

1. Show that x

x2+ y2 is harmonic.

2. Is f(z) = z3 analytic?

3. Find the invariant point of the transformation w = 1

z−2i4. Show that xy2 cannot be the real part of an analytic function.5. Find the image of x2+y2 = 4 under the transformation w=3z.6. f(z) = u + iv is such that u and v are harmonic is f(z) analytic always? Justify.7. State the Cauchy-Riemann equations in polar coordinates satisfied by an analytic

function.

8. Find the invariant points of the transformation w=2 z+6z+7 .

9. Find the analytic region of f(z) = (x-y)2+2i (x+y).10. Find the critical points of the transformation w2= (z-α )(z-β).11. Define conformal mapping.12. For what values of a,b and c the function f(z) = x – 2ay + i(bx-cy) is analytic?13. If u+iv is analytic, show that v-iu is also analytic.14. Find the image of the circle |z| = 2 under the transformation w = 3z.15. Define bilinear transformation.16. Define analytic function of a complex variable.17. Give an example such that u & v are harmonic but u+iv is not analytic.18. Find ‘a’ so that u(x,y) = ax2-y2+xy is harmonic.19. State the orthogonal property of an analytic function.

6

20. Under the transformation w = iz + I show that the half plane x¿0 maps into the half plane w¿1.

21. Find the points in the z plane at which the mapping w = z + z−1, (z≠0) fails to be conformed.

22. Prove that tan−1 yx is harmonic.

23. Show that the function f(z) = zz is not analytic at z=0.24. f(z) = r2 (cos2θ+i sin pθ) is analytic if the value of p is ….?

a) ½ b) 0 c) 2 d) 125. Define Mobius transformation.

26. If u≠ iv = 1z ; then prove that the families of curves u = c1 and v = c2 ( c1 , c2 being

constants) cut orthogonally.27. Define Isogonal transformation.28. Verify whether w = sin xcos h y+ icos x sin h y is analytic or not.29. Find the bilinear transformation which maps the points z = -2, 0, 2 into the points w = 0,

i, -i respectively.

30. If f(z) is a regular function of z, prove that ( ∂2

∂x2 + ∂2

∂ y2) |f(z)|2 = 4 |f ’(z)|2

Part B

1. Find the analytic function whose real part is sin 2x

cosh2 y−cos2 x .

2. Find the image of the infinite stepsi) ¼ < y < ½

ii) 0 < y < ½ under the transformation w = 1z

3. Find the bilinear transformation which maps -1, 0, 1 of the z-plane onto -1, -i, 1 of the w-plane. Show that under this transformation the upper half of the z-plane maps onto the interior of the unit circle |w|=1.

4. Find the analytic function f(z) = u + iv, where v = 2sin x sin h y

cos2 x+cos h2 y .

5. Find the bilinear transformation which maps the points z = 0, 1, ∞ into w=i,-1,-i.6. Prove that x2 – y2 + e−2x cos2 y is harmonic and find its harmonic conjugate.

7. If φand Ψ are functions of x and y satisfying Laplace equation namely ∂2φ∂ x2 + ∂

2φ∂ y2 = 0;

∂2Ψ∂ x2 +∂

2Ψ∂ y2 = 0 and u = ∂φ∂ y

−∂Ψ∂x ; v = ∂φ∂ x

+ ∂Ψ∂ y . show that u +iv is analytic.

7

8. Show that the function f(z) = √¿ xy∨¿¿ is not regular at the origin, through C-R equations are satisfied at origin.

9. Determine the region D’ of the w-plane into which the triangular region D enclosed by the lines x=0, y=0, x+y=1 is transformed under the transformation w = 2z.

10. Find the analytic function f(z) = u +iv if u + v = x

x2+ y2 and f(1)=1.

11. Show that the mapping w = i+ zi−z , the image of the circle x2+y2 < 1, is the entire half of

the w-plane to the right of the imaginary axis.12. Show that an analytic function with constant modulus is also constant.13. Find analytic function f(z)=u(r,θ)+iv(r,θ) such that v(r,θ)=r2cos2θ−¿rcosθ+2.

UNIT – IV(COMPLEX INTEGRATION)

PART -A

1. Evaluate ∫0

1+i

( x− y+i x2 )dz along the line from z = 0 to 1+i.

2. Evaluate ∫c

sin z dz along the line z=0 to z=i.

3. Prove that ∫c

(z−a)ndz=0, [n=-1] where c is the circle. |Z-a|=r

4. Evaluate ∫0

2+i

(Z)2dz along the line y¿x2

5. State Cauchy’s integral theorem.

6. Evaluate ∫c

dz2 z−3 where c is the circle |Z|=1.

8

7. Evaluate ∫c

dzz ez

where c is the circle |Z|=1.

8. Evaluate ∫ce

1z where c is the circle |Z|=1.

9. Define Taylor’s series.

10. Define Laurent’s series

11. Define Singularity.

12. Find the residue of f(z)= 1z2 ez

13. Find the residue of f(z) = z+1

z2(z−2) at each of the poles.

14. State Cauchy’s Residue theorem.

15. State Jordan’s Lemma

PART _ B

1. Evaluate ∫c

z2dz where the ends of c are A(1,1) and B(2,4) given that

(i)C is a curve y=x2

(ii)C is the line y=3x-2

2. Evaluate , using cauchy’s integral formula 12πi∫c

z2+5z−3

dz on the circles (i) |z|=4 and |z|=1

3. Show that ∫c

(Z+1)dz = 0 where C is the boundary of the square whose vertices are at the

point Z=0,Z=1,Z=1+I,Z=i.

4. Using Cauchy’s integral formula find the value of ∫c

(z+4)dzz2+2 z+5

where c is the circle |z+1-

i|=2

9

5. Evaluate ∫c

e2 z

( z−1 ) ( z−2 ) dz where c is the circle |z|=3

6. Evaluate ∫c

sinπ z2+cosπ z2

(z−1 ) ( z−2 ) dz where c is the circle |z|=3

7. Evaluate ∫c

3 z2+z( z2−1 ) dz where c is the circle |z-1|=1

8. Evaluate ∫c

(z+1)dzz2+2 z+4

where c is the circle |Z+1+i|=2

9. Evaluate ∫csin6 z dz

( z−π6)

3 where c is the circle |Z|=1

10. Evaluate∫c

tanz /2dz(z−a)2 ,-2<a<2 where ‘C is the boundary of the square whose sides are x=

±2 and y=±2.

11. Evaluate ∫c

1+z( z3−2 z2 )dz where c is the unit circle |z|=1

12. Expand cos z as a Taylors series about the points (i)Z=0 (ii) z= π/4

13. Expand f(z) = z2−1( z+3 ) ( z+2 )

in a Laurent’s series if (i) |z|>3 (ii) |z|<3

14. Expand 1

z2−3 z+2 when 1<|z|<2 by Lauren’s Series.

15. Obtain the Laurent’s expansion for ( z−2 )(z+2)( z+1 ) ( z+4 )

which are valid (i) 1< |z|<4 (ii) |z|>4

16. If 0<|z-1|<2, then express f(z)= z

( z−1 ) (z−3 ) in a series of positive and negative powers of

z-1.

17. Find the residue of f(z) = z2

(z−1)2 ( z+2 ) at each of the poles.

18. Find the residue of f(z) = 1

(z2+1)2 about each singularity.

10

19. Evaluate ∫c

(2 z−1)dzz ( z+1 )(z−3)

where c is the circle |z|=2

20. Evaluate ∫c

(z2−2 z)dz( z+1 )2(z2+3)

where c is the circle |z|=3 using residue theorem

21. Evaluate ∫c

z sec z dz(1−z2)

where c is the ellipse 4x2+9y2 = 9.

22. Evaluate ∫0

2π dθ13+5 sinθ

23. Show that ∫0

2π dθa+bcosθ

= 2π√a2−b2

, a>b>0

24. ∫0

2π sin2θa+bcosθ

dθ=2 πb2 [a−√a2−b2]where 0<b< a

25. Evaluate ∫0

2π dθ1−2asinθ+a2 ,0<a<1

26. Evaluate ∫0

π 1+2cosθ5+4cosθ

dθ

27. Prove that ∫0

∞ dx(x2+1)2=

π4

28. Evaluate ∫−∞

∞ x2

(x2+a2)(x2+b2)dx ,a>0 , b>0

29. Evaluate ∫−∞

∞ x2−x+2x4+10 x2+9

dx .

30. Evaluate ∫0

∞ cos ax(x2+1)

dx ,a>0

11

UNIT-V (LAPLACE TRANSFORM)

PART-A

1. Define Laplace transforms

2. Find the Laplace transform of

1−cos tt

3. Find the inverse Laplace transform of cot−1 ( ks )

4. Find L−1 {cos−1( s )}

5. Find the Laplace transform of unit step function.6. State the conditions under which Laplace transform of f(t) exists.7. State the first shifting theorem on Laplace transforms.8. State the second shifting theorem on Laplace transforms.

9. Verify initial value theorem forf ( t )= 1+e-t(sin t+cos t ).10. Find the Laplace transform of t cos at

11. Find the Laplace transform of t sin 2 t

12. Find L−1{ 1

s2+4 s+4 }13. Find L (e−3t sin t cos t )

14. Find inverse Laplace transform of e−as

s

15. Find inverse Laplace transform of e−2 s

s−3

16. If L (f ( t ))=1

s (s2+a2 ) ,find Ltt→0f ( t ) and Lt

t→∞f ( t )

17. Verify the finial value theorem for f ( t )=3e-t

12

18. Find the Laplace Transform of

f ( t )=¿{0, t < 2π3 ¿ ¿¿¿

19. Prove that L (sin at )=a2

s2+a2

20. Find L (3e5 t+5 cos t )

21. Find L (cos3 3t )

22. FindL (sin2 t cos3 t )

23. FindL(∑n=0

N

an e-bt cosnt)

24. Find L(e−7 t . t−1

2 )25. Find the Laplace Transform of f ( t )=¿ {et , 0<t <1¿ ¿¿¿26. Define change of scale property

27. Find L(sin2 t

t )28. Find

L( 1−cos tt )

29. Find L (t e-t cosht )

30. Find the laplace transform of [∫

0

t

te−t sin tdt ]31. Define convolution theorem.32. Define convolution of two functions.33. Define initial value theorem.34. Define finial value theorem

35. Find L−1 [ log(1+s

s2 )]PART-B

1.Using convolution theorem find the inverse Laplace transform of

13

1( s2+1 )(s+1)

2.Find using convolution theorem.

3.Find using Convolution theorem

4.Using Convolution theorem, find

5.Using Convolution theorem, find the inverse Laplace transform of

6.Find the Laplace transform

7.Find the Laplace transform

8.Find the Laplace transform of square wave function defined by with period 2a

9.Find the Laplace transform of the following triangular wave function given by

f ( t )=¿ { t, 0≤ t ≤π ¿ ¿¿¿and f ( t+2π )=f ( t )10.

Find the Laplace transform of the Half wave rectifier function

f ( t )=¿ {sinωt , 0< t < πω ¿ ¿¿¿

with f ( t+ 2πω )= f ( t )

11.Find the Laplace transform of square wave function given by where f ( t+a )=f ( t )

12.Verify the initial and finial value theorem for the function

13.Verify the initial and finial value theorem for the function

14.Solve the differential equation by using

Laplace transform method

14

L−1[ s(s2+a2 )2 ]

L−1[ 1s (s2+4 ) ]

L−1[ 1(s+1)( s2+1) ]

s2

( s2+a2 )(s2+b2)

with f (t+2a )= f ( t )f ( t )=¿ { t, 0< t <a ¿ ¿¿¿and f ( t+2 )=f ( t ) for t>0f ( t )=¿ { t, 0< t <1 ¿¿¿¿

f ( t )=¿ {1, 0< t <a ¿ ¿¿¿

f ( t )=¿ {E, 0<t < a2 ¿ ¿¿¿

f ( t )= 1+e-t(sin t+cos t )

f ( t )= 1-e-at

d2 ydt 2

+ y=sin 2t ; y (0)=0 and y' (0)=0

15.Solve the differential equation by

using Laplace transform method16.

using Laplace transform method solve the differential equation

y - 3y' - 4y=2e rSup { size 8{ - t} } ;with y \( 0 \) =1∧ y ' \( 0 \) =1} { ¿

17.Solve using Laplace

transform.18.

Solve the differential equation using

Laplace transform method.19.

Solve using Laplace transform.20.

Solve the differential equation using Laplace

transform method.21.

Solve using Laplace transform.22.

Find the Laplace transform of 23.

Evaluate using Laplace transform.24.

Find 25.

Solve the equation 26.

Find the Laplace transform of 27.

Find

15

d2 ydt 2

−3 dydt

+2 y=e−t ; with y (0 )=1 and y'(0)=0

d2 ydt 2

+4 dydt

+4 y=sin t ; if dydt

=0 and y=2 when t=0

y - 3y'+2y=4e rSup { size 8{2t} } ; y \( 0 \) = - 3∧ y ' \( 0 \) =5} {¿

d2 xdt2

−3 dxdt

+2x=2 ; given x=0 dxdt

=5 f or t=0

y +9y=cos 2t; y \( 0 \) =1∧ y left ( { {π} over {2} } right )=-1} {¿

d2 ydt2

+2 dydt

+5 y=e−t sin t ; y (0)=0 ,y'(0 )=1

∫0

∞

t e-2t cos t dt

eat−e−bt

t

L−1{1s

ln( s2+a2

s2+b2 )}dydt

+4 y+5∫0

1

y dt =e-twhen y (0 )=0

t e-2t cos 3t

L[cosat−cosbtt ]

28.Find the Laplace transform of

SNS COLLEGE OF ENGINEERING, CBE – 107

INTERNAL ASSESSMENT – I

(COMMON FOR ALL BRANCHES)

MATHEMATICS – II

PART – A

1. Define linear differential equations 2. Solve (4D2-4D +1)y = 4

16

e−4 t∫0

t

t sin 3 t dt

3. Solve (D2-3D -4) y = e 3x +e-x

4. Transform the equation (2x+3)2 d2 ydx2 - 2(2x+3)

dydx - 12y =6x into a linear differential

Equation with constant co-efficient

5. Find the particular integral of (D2_4)y = cosh2x6. Define gradient of the scalar functionφ.7. Find the value of a , b , c so that the vector

F⃗=( x+2 y+az ) i⃗+(bx−3 y−z ) j⃗+(4 x+cy+2 z )k⃗ is irrotational.8. If φ = log (x2+ y2+z2 ¿ find∇ φ.9. Find the unit vector normal to the surface x2− y2+z=2 at the point (1 , -1 ,2)10. Prove that curl (gradφ) = 0.

PART – B

11. (a)(i) Solve the system of equations dxdt +2x+3y =2e2t ;

dydt +3x+2y =0 (10)

(ii) (D2+9)y = 11 cos3x (6)(OR)

(b)(i) Solve ((3x+2)2d2 ydx2 +3(3x+2)

dydx -36)y =3x2+4x+1 (10)

(ii) Solve (D2-3D -4) y = e 3x +e-x (6)12. (a)(i)Solve (D2+3D+2)y = sin3x cos2x (8)

(ii)Solve(x2D2+ xD+1) y= logx sin (logx) (8) (OR)

(b)(i) Solve (4D2-4D +1) y = 4 (ii) Solve (D2+1) y = sin2x

13. (a) Solve the simultaneous differential equations dxdt +2y=sin2t ;

dydt -2x =cos2t (16)

(OR)

(b) Verify Green’s theorem for ∫C

{( x2− y2)dx+2 xydy } , where C is the boundary of

the rectangle in the XOY – plane bounded by the lines x = 0, x = a, y = 0 and y = a.

14. (a)(i) If r⃗=x i⃗+ y j⃗+z k⃗ prove that (i) ∇ r= r⃗r , (ii) ∇ rn=nrn−2 r⃗ ,where r=¿ r⃗∨¿¿.

(ii) Prove that ∇ ∙ ( F⃗× G⃗ )=G⃗ ∙ (∇×F⃗ ) – F⃗ ∙ (∇×G⃗ )(OR)

17

(b) Verify the Divergence theorem for F⃗=(x2− yz ) i⃗+( y¿¿2−zx) j⃗+(z2−xy) k⃗ ¿ taken over the rectangular parallelepiped 0 ≤ x ≤ a ,0 ≤ y ≤ b , 0 ≤ z ≤ c .

15. (a) Verify Stoke’s theorem for a vector defined by F⃗=(x2− y2) i⃗+2 xy j⃗ in the rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.

(OR) (b) Verify Stoke’s theorem for a vector defined by F⃗= y i⃗+z j⃗+x k⃗ , where S is the upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.

18