Department of Mathematics

Birla Institute of Technology, Mesra, Ranchi

MA 2201(Advanced Engg. Mathematics) Session: 2017----

Tutorial Sheet No. -- 2

MODULE – IV

1. Reduce the following linear homogeneous differential equations into the Strum Liouville

form:

(i) .00)1( ≠=+′−+′′ xnyyxyx

(ii) .10)1( 22 <=+′−′′− xynyxyx

(iii) .10)1(2)1( 2 <=++′−′′− xynnyxyx

(iv) 022 =+′−′′ nyyxy

2. Find the eigen-values and eigen-functions of the following Strum Liouville boundary

value problem. Verify the orthogonality by direct calculations.

(i) 0)(,0)0(,0 ===+′′ πλ yyyy

(ii) 0)(,0)0(,0 =′=′=+′′ lyyyy λ

(iii) ( ) )2()0(,2)0(,0 ππλ yyyyyy ′=′==+′′

3. State Dirichlet’s conditions for the expansion of any function )(xf in terms of Fourier

series. If )(xf is a periodic function of period 2π and it is expressed in the series form

as:

∑∑∞

=

∞

=

++=11

0 sincos2

)(n

n

n

n nxbnxaa

xf , π2+<< lxl , then show that

.sin)(1

,cos)(1

,)(1

222

0 ∫∫∫+++

===πππ

πππ

l

l

n

l

l

n

l

l

nxdxxfbnxdxxfadxxfa

4. Find the Fourier series for the function f(x) = (π –x)2/4 in ( 0, 2π ).

Hence deduce that (i) 6

..................4

1

3

1

2

1

1

1 2

2222

π=

++++ ,

(ii) 12

..................4

1

3

1

2

1

1

1 2

2222

π=

+−+− .

5. Find the Fourier series to represent the function f(x) = x2 in (-π, π ).

Hence deduce that (i) 6

..................4

1

3

1

2

1

1

1 2

2222

π=

++++ ,

(ii) 12

..................4

1

3

1

2

1

1

1 2

2222

π=

+−+− ,

(iii) 8

.................5

1

3

1

1

1 2

222

π=

+++

6. Show that the Fourier series to represent the function f(x) = x sinx in (-π, π ) is:

−+−−−= ..................

5.3

4cos

4.2

3cos

3.1

2cos2cos

2

11sin

xxxxxx

Hence deduce that .............................................................7.5

1

5.3

1

3.1

1

2

1

4−+−+=

π

7. Assume an alternating current after passing through a rectifier posses the form

≤≤

≤≤=

ππ

π

2,0

0,sin0

x

xxII

where I0 is the maximum current. Express I as Fourier series.

8. Find the Fourier series to represent the function f(x) = x-x2 in (-1, 1 ) as

,sin)1(2

cos)1(4

2

1)(

1122 ∑∑

∞

=

∞

=

−−

−−=

n

n

n

n

n

xnxn

nxf

π

ππ

π

Hence deduce that ..................5

1

3

1

1

1

8 222

2

+++=

π

9. Assume a sinusoidal voltage E sinωt which is passed through a half wave rectifier and

clips the negative portion of the wave. Express the resulting periodic function

<<

<<−=

2/0,sin

02/,0)(

TttE

tTtU

ω where T = 2π /ω, in a Fourier series as:

+++−+= ..................

7.5

6cos

5.3

4cos

3.1

2cos2sin

2)(

tttEt

EEtU

ωωω

πω

π

10. Find the Fourier series to represent the function defined as

<<

<<=

21,0

10,)(

x

xxxf

11. Find the half-range sine and cosine series for the function ex

in 0 ≤ x ≤ 1.

12. Obtain the half-range cosine series for the function f (x) = (1-x)2 in the interval 0≤ x ≤ 1

as

++++= ..................

3

3cos

2

2cos

1

cos4

3

1)(

222

xxxxf

πππ

π

13. Obtain the half-range sine and cosine series for the function

≤≤−

≤≤=

axa

axxf

2/,1

2/0,1)(

14. Show that the series ∑∞

=1

2sin

11

n l

xn

n

π

π represents x

l−

2when .0 lx <<

15. If

≤≤−

≤≤=

πππ

π

xxm

xmxxf

2/),(

2/0,)(

then obtain the half-range sine series for the function as

−+−= ..................

5

5sin

3

3sin

1

sin4)(

222

xxxmxf

π

MODULE – V

16. Derive Cauchy Riemann partial differential equations for the necessary conditions of

analyticity of a function of complex variable. When these conditions become sufficient?

17. Show that continuity does not imply differentiability by considering the function

.)(2

zzf =

18. Verify whether the function

=

≠+

−

=

0,0

0,)(

)( 26

3

z

zyx

ixyyx

zf is non - analytic at 0=z .

19. If ivuzf +=)( is an analytic function of z, prove that the two curves 1),( cyxu = and

2),( cyxv = will intersect orthogonally.

20. Derive Cauchy Reimann equations in polar form and prove that

011

2

2

22

2

=∂

∂+

∂

∂+

∂

∂

θ

u

rr

u

rr

u and 0

112

2

22

2

=∂

∂+

∂

∂+

∂

∂

θ

v

rr

v

rr

v

21. If φ and ψ are functions satisfying Laplace’s equation, show that S+iT is analytic, where

xy

S∂

∂−

∂

∂=

ψφ and

yxT

∂

∂+

∂

∂=

ψφ

22. Determine p such that the function

++= −

y

pxiyxzf

122tan)log(

2

1)( is an analytic

function.

23. If ψφω i+= represents the complex potential for an electric field and

22

22

yx

xyx

++−=ψ , determine the functionφ .

24. State and prove Cauchy’s Integral Theorem.

25. State and prove Cauchy’s Integral Formula.

26. Evaluate ∫+

=i

dzzI

2

0

2

1 along the line OA where A is the point (2+i).

27. Evaluate dzzc

∫ , where C is the contour

i) straight line AB from iiz to−=

ii) left half of the unit circle 1=z .

28. Evaluate dzz

e

C

z

∫−

2round the contour C, where C is the circle 1=z .

29. Use Cauchy’s integral formula to evaluate ∫ −

+

Cz

zz

1

32

2

where C is the circle 2=z .

30. Evaluate the integrals

i) ∫−C

dzz

z

1

cosπ ii) ∫

−−C

dzzz

z

)2)(1(

cos 2π (iii) dz

z

e

C

z

∫ + 4

2

)1(

where C is the circle 2=z .

MODULE – VI

31. Find the image of 12 =− iz under the mappingz

1=ω .

32. Show that the image of the hyperbola 122 =− yx under the transformation z

w1

= is the

Lemniscate φρ 22 Cos= .

33. Show that under the transformation iz

izw

+

−= real axis in the z-plane is mapped into the

circle 1=w . What portion of the z-plane corresponds to the interior of the circle?

34. Show that the transformation 4

32

−

+=

z

zw changes the circle 0422 =−+ xyx into the

straight line 034 =+u .

35. Find the bilinear transformation which maps the points z=0,-i,-1 into w=i, 1, 0

respectively.

36. Expand )3)(1(

1)(

++=

zzzf in Laurent’s series valid for the regions

(i) 1< z <3 (ii) 0 < 21 <+z .

37. Find the Taylor’s or Laurent’s series which represent the function )3)(2(

1)(

2

++

−=

zz

zzf

when (i) 2 < 3<z , (ii) 2<z , (iii) 3>z .

38. For the following functions, find the poles and the residues at each poles:

(i) )3)(2(

1)(

2

++

−=

zz

zzf , (ii)

)3)(2()1()(

4

2

−−−=

zzz

zzf (iii)

2)(

cot)(

az

nzzf

−=

39. State Cauchy’s theorem of residues. Evaluate the following integrals using Cauchy’s

theorem of residues:

dzzz

z

C

∫ −+

+

)2()1(

122

2

where C: (a) 3=z , (b) 11 =−z

40. Use the method of contour integration to prove that:

(i) 3

2

cos2

12

0

πθ

θ

π

=+∫ d , (ii) 10,

1

2

cos21

12

2

0

2<<

−=

−+∫ aa

daa

πθ

θ

π

,

41. Use the method of contour integration to prove that:

(i) 2)1(

12

π=

+∫∞

∞−

dxx

(ii)

−

−=

++

−−∞

∞−

∫ a

e

b

e

badx

bxax

x ab

222222 ))((

cos π

MODULE –VII

42. Form the partial differential equations for the following:

(i) f (x2 +y

2 - z

2, x+y+z )=0 , (ii) 2z =

2

2

2

2

b

y

a

x+ , (iii) )()( 22 yxgyxfz −++= .

43. Solve the following differential equations:

(i) xy2 p+xzq = y

2, (ii) )()()( 22 yxqyzxpzxy −=+−+ ,

(iii) (x2-yz)p+(y

2-zx)q = z

2-xy,

(iv) x2(y-z)p+y

2(z-x)q=z

2 (x-y), (v) ( y

2+z

2-x

2)p -2xyq+2zx = 0.

44. Solve the following partial differential solutions,y

Dx

D∂

∂≡′

∂

∂≡ and where

(i) xyzDDDD 12)152( 22 =′−′−

(ii) )32sin()2( 22 yxzDDDD +=′−′−

(iii) yxezDDD x 222 32)2( −=′−

(iv) xeyzDDDD )1()2( 22 −=′+′−

(v) )2log(16)44( 22 yxzDDDD +=′+′−

45. Solve the following partial differential solutions,y

Dx

D∂

∂≡′

∂

∂≡ and where

(i) )2cos()1( 2 yxzDDDD +=−′+′−

(ii) )3tan(2)23( 22 xyezDD x +=−′−

46. A string is stretched and fastened to two point’s l cm. apart. The string is displaced and

then released to vibrate in the x-t plane, where ),( txu denotes the vertical displacement

of the vibrating string. The initial- boundary value problem modeling the motion of the

string is given by 0,0,2

22

2

2

><<∂

∂=

∂

∂tlx

x

uc

t

u, with the boundary conditions:

0,0),(,0),0( >== ttlutu , and the initial

conditions: .0)0,(),()0,( =−= xuxlxxu tµ Solve it by the method of separation of

variables.

47. Solve the one dimensional wave equation 0,0,2

22

2

2

><<∂

∂=

∂

∂tlx

x

uc

t

u

with the boundary conditions: 0,0),(,0),0( >== ttlutu , and the initial conditions:

)./(sin)0,(,0)0,( 3 lxbxuxut

π==

48. Solve the one dimensional heat conduction equation

2

22

x

uc

t

u

∂

∂=

∂

∂ 0,0 ><< tlx

with the boundary conditions: 0,0),(,0),0( >== ttlutu , and the initial conditions:

.0,sin3)0,( lxl

xxu <<

=

π

49. Solve the following boundary value problem:

2

22

x

uc

t

u

∂

∂=

∂

∂, 0,0 ><< tlx ,

subject to the condition 0),(

,0),0(

=∂

∂=

∂

∂

x

tlu

x

tuand .0,)0,( lxxxu <<=

50. Solve the following problem:

2

2

x

u

t

u

∂

∂=

∂

∂, for the conduction of heat along a rod without radiation, subject to the

following conditions:

(i) 0),(

,0),0(

=∂

∂=

∂

∂

x

tlu

x

tu

(ii) .0,)0,( 2 lxxlxxu <<−=