mathematics 2

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MA2261 MATHEMATICS II

UNIT I – Ordinary Differential EquationHigher order linear differential equations with constant coefficients – Method of variation of parameters – Cauchy’s and Legendre’s linear equations – Simultaneous first order linear equations with constantcoefficients.

1.Solve the following differentaial equations :

(a ) d2 y

dx 2− 3

dydx

− 4 y = 0

(b )d3 ydx3

− 3d2 ydx2

+ 3dydx

− y = 0

(c ) d 4 ydx 4

− 5d2 ydx2

+ 4 y = 0

(d )d4 y

dx4+ 8

d2 y

dx2+ 16 y = 0

(e )d3 ydx3

+ 6d2 ydx2

+ 11dydx

+6 y = 0

( f )d2 y

dx2+ 4

dydx

+29 y = 0 , given when x = 0 , y = 0 anddydx

= 1 . 5

2.Solve the following differential equations :

(a )d3 y

dx3+ y = 3+5ex

(b )d2 y

dx2− 4 y = ( 1+ex )2

(c )d2 ydx 2

+ 4dydx

+5 y = −2 cosh x

3. Solve the following differential equations :(a ) (D−2 )2 y = 8 (e2 x+sin 2 x )(b )( D2−4 D+3 ) y = sin 3 x cos 2 x

(c )( D3+1) y = sin 3x − cos 2( x2 )

4 . Solve the following differential equations :(a ) (D−2 )2 y = 8x2

(b )d2 y

dx2+

dydx

= x2+2 x+4

(c )d2 ydx 2

+ y = e2 x+ cosh 2 x + x3

Dr. D. Saravanan, Professor of Mathematics

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5. Solve the following differential equations :

(a ) d2 y

dx2− 4 y = x sinh x

(b ) d 4 ydx 4

− y = cos x cosh x

(c ) (D2−2 D ) y = ex sin x(d ) ( D2 + 4 D +8) y = 12 e−2 x sin x sin 3 x

(e ) d2 y

dx 2+ 2 y = x2e3 x+ ex cos 2 x

6 . Solve the following differential equations :

(a )d2 y

dx2− 2

dydx

+ y = x ex sin x

(b ) ( D2 − 4 D + 4 ) y = 8 x2 e2 x sin 2 x

(c )d2 y

dx 2+ 4 y = x sin x

7 . Solve the following differential equations by var iation of parameters :

(a )d2 y

dx2+ 4 y = 4 sec2 (2 x )

(b ) d2 ydx 2

+ y = cosec x

(c ) d2 y

dx2+ y = sec x

(d ) d2 ydx2

+ 4 y = tan 2 x

(e ) d2 ydx 2

+ y = x sin x

8 . solve the following Cauchy ' s hom ogeneous linear differential equations :

(a ) x3d3 ydx 3

+ 2 x2d2 ydx2

+ 2 y = 10 (x+1x )

(b ) x2d2 y

dx2− x

dydx

− 3 y = x2 log x

(c ) x2d2 ydx 2

+ xdydx

+ y = log x sin ( log x )

(d ) x2 d2 ydx2

− 3 xdydx

+ 5 y = sin ( log x )

(e ) x2 d2 y

dx2+ 2 x

dydx

− 12 y = x3 log x


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9 . Solve the following Legendre ' s linear differential equations :

(a ) (3 x+2 )2d2 y

dx 2+ 3 (3 x+2 ) dy

dx− 36 y = 3 x2+4 x+1

(b ) (2 x+1 )2d2 ydx2

− 6 (2 x+1 )dydx

+ 16 y = 8(1+2 x )2

(c ) (2 x+3 )2d2 ydx2

− 2 (2 x+3 ) dydx

− 12 y = 6 x

10. Solve the following simul tan eous differential equations :

(a )dxdt

+ 4 x + 3 y = 1 ;dydt

+ 2 x + 5 y = e t

(b )dxdt

= 7 x − y ;dydt

= 2 x + 5 y

(c ) dxdt

+ y = sin t ;dydt

+ x = cos t ; given x=2 , y=0 when t=0

UNIT II - VECTOR CALCULUS

Gradient Divergence and Curl – Directional derivative – Irrotational and solenoidal vector fields – Vector integration – Green’s theorem in a plane, Gauss divergence theorem and stokes’ theorem (excluding proofs) – Simple applications involving cubes and rectangular parallelpipeds.

Grad, Curl, Divergence

1. Find the unit normal vector at the point (2, -2, 3) to the surface x2y + 2xz = 42. Find the unit normal vector to the surface x2y + 2xz2 = 8 at (1, 0, 2)3. Find the directional derivative of = x2yz + 4xz2 + xyz at (1, 2, 3) in the

direction 2i + j – k4. Find the maximum directional derivative of = 3x2 + 2y - 3z at (1, 4, 1) in the

direction 2i + 2j – k5. Find the angle between the surfaces x2 + y2 + z2 = 9 and z = x2 + y2 - 3 at

(2, 1, -2).6. Find the angle between the surfaces x2 + y2 = 4 – 5z and x2 + y2 + 3z2 = 104

at (5, 2, -5).7. Find the angle between the normals at (1, 1, 1) and (4, 1, 2) to the surface xy

– z2 = 0.8. Find the angle between the normals at (3, 3, -3) and (4, 1, 2) to the surface

xy = z2.9. Find the angle between the tangent planes to the surfaces x log z = y2 - 1, x2

y = 2 – z at (1, 1, 1).10. For what value of ‘ k ‘ is the vector rk r solenoidal.


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11. Find ‘m’ , if (2 x+ y ) i + (4 x−11 y+3 z ) j + (3 x+mz )k is solenoidal.

12. Determine f(r) so that f (r ) r is solenoidal

13. Show that F = (2xy+z3 ) i + x2 j + 3 xz { k ¿ is conservative. Find such that = F.

14. Show that F = ( y2 cos x+z3 ) i + (2 y sin x−4 ) j + 3 xz2 k is irrotational. Also find the scalar potential.

15. Prove that ∇( 1

r2) =−2 R

r 4

16. Prove that ∇( 1

rr ) =−2 r

r3

17. Prove that ∇(r n) = nr n−2 r18. Prove that div (rnR) = (n+3)rn

19. Prove that curl (rnR) = 020. Prove that div(curl F ) = 021. Prove that curl (grad ) = 022. Prove that curl curl F = grad div F - 2F

23. If F = x2 y i − 2 xz { j + 2 yz { k ¿¿ , find curl curl F.

24. Show that ∇2 ( f (r )) = f ' ' (r ) + 2

rf ' (r )

25. ∇ .(ϕ ∇ ψ − ψ ∇ ϕ )= ϕ ∇2ψ − ψ ∇ 2 ϕ

Line Integral

26. Evaluate

∫ F . d r from (0 , 0 , 0 ) to (1 , 1 , 1) along x = t , y = t2 , z = t3 and F=(3 x2+6 y ) i − 14 yz { j + 20 xz 2 k ¿

27. Evaluate the line integral ∫c

( x2+xy )dx + ( x2+ y2 )dy where C is the square

formed by the lines y = 1 and x = 1.

28. Evaluate the work done by the force F = (5 xy−6 x2 ) i +( 2 y−4 x ) j along the curve C in the xy-plane, y = x3 from (1, 1) to (2, 8).

29. F = 5 xy { i + 2 y j ¿ Evaluate ∫c

F . d r where C is the part of the curve y = x3 between

x = 1 and x = 2

30. Prove that ∫c

F . d r is independent of path where

F = ( y2 cos x + z3 ) i +( 2 y sin x−4 ) j + 3 xz2 k Also evaluate the integral from (0, 0, 0) to (/2, 0. 1)


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Surface Integral

31. Evaluate ∫( F . n )d s

where F = 6 z i − 4 j + y k and S is the portion of the plane 2x + 3y +6z =12 in the first octant.

32. F = y i + 2 j + xz { k m /sec .¿ Show that the flux of water through the parabolic cylinder y = x2, 0 x 3, 0 z 2 is 69 m3/sec.

33. Evaluate ∬

s

Curl F . n ds where F = ( x2− y2) i + 2 xy { j ¿ over the surface of the

rectangle in the plane z = 0 and bdd by the line x = 0, y = 0, x = a, y = b.

34. F = y i + z j + xz { k m /sec .¿ Show that the flux of water through the parabolic cylinder y = x2, 0 x 3, 0 z 2 is 75 m3/sec.

Green’s Theorem

35. Evaluate ∫c

( xydx −x2dy by converting this into a double integral. It is given

that C is the bounding of the region bdd byy = x and x2 = y.

36. Prove that area bdd by simple closed curve is given by

12∫( xdy − ydx ) .

Hence find the area bounded by the parabola y2 = 4ax and its latus rectum.

37. Find the area bounded by the ellipse

x2

a2+ y2

b2= 1 .

38. Using Green’s Theorem in plane, evaluate ∫c

(( y − sin x )dx + cos x dy ) where C

is the triangle OAB whose vertices are O(0, 0), A((/2, 0), B(/2, 1)

39. Apply Green’s Theorem in the plane to evaluate ∫c

(3 x2 − 8 y2 )dx + ( 4 y − 6 xy ) dy

where C is the boundary of the region defined by x = 0, y = 0 and x + y = 1

Stokes Theorem

40. Verify Stokes Theorem for F = ( y−z+2) i +( yz+4 ) j − xz { k ¿ where S is the surface of the cube x = 0, x = 2, y = 0, y = 2, z = 0, z = 2 above the XOY plane.

41. Verify Stokes Theorem for the region z = 0 plane bounded by x = 0, x = a, y =

0 and y = b for F = ( x2− y2 ) i + 2xy { j . ¿


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42. Verify Stokes Theorem for A =(2 x− y ) i − yz2 j − y2 z k . where S is the upper half of the sphere x2 + y2 + z2 = 1 and C is its boundary.

43. Verify Stokes Theorem for A = y i +2 yz { j+ y2 k . ¿ taken over the upper half of surface S of the sphere x2 + y2 + z2 = 1, z 0 and the bounding circle x 2 + y 2 = b, z = 0 .

44. A = y i + z j +x k upper half of sphere x2 + y2 + z2 = 1 and the boundary C of the circle x2 + y2 + z2 = 1, z = 0

Gauss – Divergence Theorem

45. Verify GDT for F = x2 i + y2 j +z2 k taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

46. Verify GDT for F = ( x2− yz ) i +( y2−zx ) j +(z2−xy ) k taken over the surface S bounded by x = 0, x = a, y = 0, y = b, z = 0, z = c.

47. Using GDT evaluate ∫s

F . n ds where F = 4 xz { i − y2 j + yz { k ¿ ¿ and S is the surface

of the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

48. Evaluate ∫s

F . d s where F = 4 x i −2 y2 j+z2 k and S is the surface bounded by

the region x2 + y2 = 4, z = 0 and z = 3.

49. If S is any closed surface enclosing Volume V and F = x i +2 y j +3 z k then

show that ∬ F . n ds= 60

UNIT III - ANALYTIC FUNCTIONS

Functions of a complex variable – Analytic functions – Necessary conditions, Cauchy – Riemann equation and Sufficient conditions (excluding proofs) – Harmonic and orthogonal properties of analytic function – Harmonic conjugate – Construction of analytic functions – Conformal mapping : w= z+c, cz, 1/z, and bilinear transformation.

1. Find the analytic region of f(z)=(x-y)2 +2i (x+y).

2. Examine whether ez is analytic.

3. Examine whether |z|2 is analytic.

4. Examine whether z2 & z3 are analytic.

5. Properties of analytic function.


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6. If f(z) is analytic prove that ( ∂2

∂ x2+ ∂2

∂ y2 )|f ( z )|2=4|f ' ( z )|2 .

7. If u=ex [ xcos y− y sin y ] ,find f (z).

8. If u( x , y )=3 x2 y+2 x2− y3−2 y2,find v, f (z).

9. Determine f (z) if v=sin 2x

cosh 2 y−cos2 x.

10. Find f (z) if v=3 x2 y− y3

11. Find f(z) if u=log √x2+ y2

12. Verify v=( xcos y− y sin y ) ex is harmonic. Construct analytic function.

13. If u−v=ex (cos y−sin y ) . find f (z).

14. If u−v= (x− y ) ( x2+4 xy+ y2 ) find f(z).

15. If 2 u+v=e2 x [ (2 x+ y ) cos2 y+( x−2 y )sin 2 y ] .Find f (z).

16. If 2 u+v=ex (cos y−sin y ) ., find f (z).

17. If f (z) is regular prove that ( ∂2

∂ x2+ ∂2

∂ y2 )|Re f ( z )|2=2|f '( z )|2 .

18. If u & v are harmonic prove that ( ∂u∂ y

−∂ u∂ x )+i(∂ u

∂ x+ ∂ u∂ y )

is analytic.

19. Give an example that both u & v are harmonic but f (z) is not analytic.

20. If f(z) is analytic in the complex plane prove that ( ∂2

∂ x2+ ∂2

∂ y2 ) log|f ( z )|=0 .

21. Find the analytic function whose imaginary part is ex2−2 y2

sin(2 xy ).

22. If v=log ( x2+ y2) is harmonic. Find the Real part of analytic function with this

function v as its Imaginary part.

23. Prove that function with constant modulus is constant.

24. Prove that analytic function with constant real part is constant.


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25. Stream functionψ=tan−1 ( y

x ). Find velocity potential.

26. If velocity potential =3x2y-y3. Find Stream function.

CONFORMAL MAPPING

27. Find the image of the y-axis under the transformationw=z2.

28. Discuss the transformationw=1

z .

29.Draw image of square (0,0) (1,0) (1,1) &(0,1) under w =(1+i)z.

30. Find the image of |z−3 i|=3 under the mappingw=1

z .

31. Image of |z+2 i|=2 underw=1

z .

32. Find the image if |z|=2 under w=z+3+2i

33. Find the image of |z−2 i|=2 ifw=1

z .

34. Image of x2− y2=1 , is ρ2=cos2 φ under

w=1z .

35. Image of x=k underw=1

z .

FIXED POINTS, CRITICAL POINTS

36. Find the critical points of w=z2

37. Find the fixed points ofw= z+4

z+1 .

38. Find the fixed points of w= z−1−i

z+2

39. Find the critical points of w=z4−4 z

40. Find the critical points ofw=( z−α ) ( z−β )


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BILINEAR TRANSFORMATION

41. Find the bilinear transformation which maps the points (-i, 0, i) onto the points (-1, i,

1) respectively

42. Find the bilinear transformation which maps the points z=1,i,-1 onto i,0,-i

43. Find the bilinear transformation which maps the points (-1,i,1) onto (1,i,-1).

44. Find the bilinear transformation which maps the points (1,i,-1) onto (0,1,).

45. Show that this transformation maps interior of unit circle to upper half w plane.

46. Find the bilinear transformation which maps the points (-1, 0, 1) to (0, i, 3i).

47. Given (-1,0,1) maps to (-1,-i,1). Show that this maps the upper half of the z-plane

onto interior of unit circle |w|=1

48. Discuss the invariant points of this transformation, (0, 1,) to (-5,-1,-3)

UNIT IV - COMPLEX INTEGRATION

Complex integration – Statement and applications of Cauchy’s integral theorem and Cauchy’s integral formula – Taylor and Laurent expansions – Singular points – Residues – Residue theorem – Application of residue theorem to evaluate real integrals – Unit circle and semi-circular contour(excluding poles on boundaries).

Evaluate the following Using Cauchy integral formula

1. where C : |z|=1

2. where C |z| =3

3. where C |z| =3

4. where |z| = 1


∫c

e2 z

z−3dz

∫c

cos πz2

( z−2 )( z−5 )dz

∫c

cos πz2+sin πz2

(z−1)2( z−2)dz

∫c

z2+1( z−2 )( z−3 )

dz

1( z+1) z

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5. around |z-i| = 2

6. where |z| =1

7. where |z| = 2

8. whereC is |z| = 1

9. where Cis |z| =1

10. C : |z-1|=3

11. Find f(z) & f’(z) Where C: |z| = 2.5

Taylor’s & Laurent Series

12. about i) z= 0 ii) z=1

13. in |z| =1

14. in |z+1| < 1

15. in |z| <1 & |z| >4

16. in |z|>2 & 0<|z-1| <1

17. in |z|<2, 2< |z|< 3, |z| > 3


∫c

ez

z−πdz

z2−4( z+1)( z+4 )

∫c

ez dz

∫c

cos πzz−1

dz

∫c

dzz−1

∫c

z+1z (2 z+1 )

dz

∫c

ez

( z+1)2dz

f (a )=∫c

2 z2−z−2( z+a)

dz

z−1z+1

1( z+1)( z+2)

z2−1( z2+5 z+6 )

1( z−1 )(z−2)

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18. in 3 < |z+2| < 5

19. in 1 < | z+1 | < 3

20. in the nbd of singular point

singular points and zeros

21.sin (1/z)

22.sin(1/(1-z))

23. (tan z)/z

24.cos (/z)

25.ez

26. .

27.

Residue Theorem

Z=a is a simple pole z lim a (z-a) f(z)

If z=a is a pole of order m

Coefficient of (z-a)-1 in the expansion of f(z) around

an isolated singularity

28. f(z) =

z

( z−1 )2( z+2) at the singular point z=1

29. where C is |z| = 4


z2−6 z−1( z−1 )(z−3)( z+2)

ez

( z−1 )2

z3−1z3+1

cos πz

( z−a )2

7 z−2( z+1) z (z−2)

z lim adm−1

dzm−1( z−a )m f ( z )

∫c

1

( z2+a)2dz

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30.∫c

z+7

z2+2 z+5dz

where C: |z-i| = 1.5

31.∫c

zsec z

1−z2dz

where C: 4x2 + 9y 2 = 36

32.∫c

zsec z

(1−z )2dz

C: |z| = 3

33.∫c

cos πz2+sin πz2

(z−1)2( z−2)dz

where C: |z| =3

34.∫c

z+4

z2+2 z+5dz

where C : |z +1-i| = 2

35.∫ e2 z

( z+1)4dz

where C : |z| =2

36.∫c

12 z−7

( z−1 )2(2 z+3)dz

where C: |z| = 2

37.∫ dz

( z2+4 )2where C : | z –i| = 2

38.∫ zdz

( z−1 )( z−2)2 C : |z-2| = ½

39.∫c

ez

z2+1 dz C: |z| = 2


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UNIT V - LAPLACE TRANSFORMATION

Laplace transform – Conditions for existence – Transform of elementary functions – Basic properties – Transform of derivatives and integrals – Transform of unit step function and impulse functions – Transform of periodic functions. Definition of Inverse Laplace transform as contour integral – Convolution theorem (excluding proof) – Initial and Final value theorems – Solution of linear ODE of second order with constant coefficients using Laplace transformation techniques.

Part A

1. L(t n) exists if n > ------- s> -------------

2. L(K) is ----------

3. Define periodic function and state its Laplace transform formula.

4. Find L (sin (2t + 3 ).

5. True or false: L{(f(t)} = (s) then L{f( t / 2 )} = 2 (2s).

6. If f(t) is a periodic function of period 2, then L{f ( t)} =--------------------

7. L{ t 3/2 + cos t + 1}

8. State the first shifting theorem on Laplace transform

9. Give two examples of functions which have no L.T

10. The period of | sin t | is………….

11. L(f(t) = 1 / (s-2) 2 then tlt

0 f(t) = ----

12. State the condition for which the L.T of f(x) exist

13. If L (f(t)) = (s) then L (f ”(t))------

14. L({et – e-3t}/t)

15. If L{(f(t)} = F(s) then show that L{f(at)} = 1/a F(s/a)

16. L {∫0

∞

te−3 t sin 2 tdt}

17. L ( t/e t)


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18. L{S an e-bt cos nt }

19. If f(t) = e -2t sin 2t find L (f ’(t))

20. Write the value of 1 * e t

21. Write the value of t * e t

22. If f(t) = t ½ find L(f(t))

23. L {0∫t e –t dt }

24. If L (f(t)) = 1/ s(s+2) then verify the initial & final value theorems

25. Give an example of a function which has L.T but it is not continuous

26. Another name of the unit step function is ----------

27. L(f(t) = 1 / (s-2) 2 then tlt

0 f(t) = ----

28. L-1( 2/ s 2)

29. If L-1((s))= f ( r ) then L -1

30. L-1 ( 1/ (s + a ) is valid for…………..

31. L-1{cot -1 ( 2/(s+1)}

32. L -1(1) = ------------

Part B

Find Laplace transform of the following

1. t e-2t sin3t

2.

cos2 t−cos3 tt

3. cos 32t

4.

sin wt 0<t< πw

0πw

<t<2 πw

5. e –t ∫

sin tt

6. Prove that L[f"(t) ]=s2Lf(t)-sf (0) – f ' (0).

7. sinh 3t cos2 t

8. t2 e -3t sin 2t

9. t sin t

10. (sin at) / t

11.


φ( s )s

f ( t )={ t 0<t<a2 a −t a< t <2 a

and f ( t +2a )=f ( t )

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12. L (f(t)=(s) then L( f(t) / t 2 ) =-----

13. Write the value of t * e t

14. (e at - cos bt) / t

15.

16. (e -3t sin 2t )/t

17. sinh 2 2t

18. e –t tn

19. A full sine wave rectifier given by f(t) = E sin wt in 0 < t < /w and f(t + /w) = f(t)

20. cos 3 2t

21. (cos 2t – cos 3t) / t

22.f ( t )={ t 0<t <2

4−t 2<t <4and f ( t+4 )=f ( t )

23.

24. (e-t sin t)/t

25.

[ t∫0

t

e−4 t cos3 tdt ]+sin 5 tt

26.

Find Inverse

27. L -1 {(s + 1)/ .(s2 + s + 1) }

28. L -1 {(e-3s)/(s-2)4 }

29. L -1 {s/((s2 + 1)(s2 + 4))}

30. L -1 {1/ ( s 2 + 4s + 4 )}

31. L-1 {(5s + 3)/(s 2 + 2s+ 5)}

32. L -1 {(2s + 3)/ (2s2 + 6s + 13)2}

33. L-1 {(1/ s4 – 1)}


f ( t )={ sin t 0< t<π2 π −t π< t <2 π

andf ( t +2π )=f ( t )

f ( t )= { t 0< t<π2 π −t π<t <2 π

and f ( t+2 π )=f ( t ) is1

s2tanh

πs2

f ( t )={ E 0≤t≤T /2−E T /2< t<T

and f ( t+T )=f ( t )

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34. L-1 {(e –s/ s2)}

35. L-1 {(3s+2)/(3s2 + 4s + 3)2}

36. If L ( ∫ f(t))=(s) then L f(t)dt =(s)/s Use this result to find L-1{1/s(s+a)}

37. L-1 ((s)) = f(t) , find L -1 (e-as (s))

38. L-1 {cot -1 ((s+3)/2)}

39. L -1 { s/(s+2)3}

40. L-1 { s / (s 2 +1)( s 2 +4)}

41. L-1{ e -2s/ s(s+1)}

42. L-1{s2 / (s2 + a2)2}

43. L-1 {(1/ s4 + 4a4)}

Using Convolution theorem

44. State and Prove the convolution theorem

45. L -1 s/(s2+1)2

46. L -1 1/(s2+4)2

47. L -1 1/ ( s2 + a2)2

48. L -1 s2 / (s2 +a2)( s2+b2)

49. L-1{ 1/ ((s+1)(s+9)2)}

50. L-1 { 1/(s+1)(s+2)}

51. L-1 { 1/(s3(s+5)}

52. L-1{ 1/ s2(s2 +25)}

53. L -1 s / (s2 +9)( s2+125)

Application of L.T.- Solve the following differential equations

54. y”+4y=sin2t, given y (0)= y’(0)= 0.

55. y” – 2 y’ + 2y = 0 y = y’ = 1 at x = 0

56. y” -2y’ + x = e –t x( 0) = 2 x’(0) =1

57. y”–y’-2y = 20 sin 2t given y(0) = 0 y’(0) = 2

58. y” + 9 y = 18 t given y(0) = 0 = y(/2)

59. y” + 4y = f(t) with y(0) = 0 and y’(0) = 1 and f(t) = t if 0 < x < 1

and 0 else where.

60. y” + y’ = t2 + 2t given y = 4 and y = -2 at t = 0

61. y” – 3y’ + 2y = e –t given y(0) = 1 & y’(0) = 0


17

62. y’’ + 2y’ -5y = e-t sin t given y(0) = 0 and y’(0) = 1

63. y” + y’ -2y = 3 cos 3t – 11 sin 3t given y(0) = 0 y’() = 6

64. y” + 2y’ + y = t e-t given y’(0) = -2

65. y” + 6y’ + 9y = 2 e-3t given y(0) = 1 & y’(0) = -2

66. y” + 4y’ +4y = t e-t given y(0) = 0 ,y’(0) = -1

67. y” + 3y’ + 2y = 2( t2 + t + 1) given y(0) = 2 y’(0) = 0

68.

69. Evaluate integral ∫ e-2t sin3t dt

70. Evaluate ∫ e-t cos 2t dt

71. Solve: y (t) = t 2 + y(x) ∫ sin ( t-x) dx


∫0

∞e−t−e−2t

tdt

mathematics 2

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