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PART - A

1) Find the partial differential equation of the family of spheres having their centres on the line x = y = z.

2) Obtain partial differential equation by eliminating arbitrary constants a and

b from .1)()(222 zbyax

3) Form the partial differential equation by eliminating a and b from

))(( 2222 byaxz .

4) Form the partial differential equation by eliminating the constants a and b

from .nn byaxz

5) Find the partial differential equation of all planes passing through the origin. 6) Find the partial differential equation of all planes having equal intercepts on

the x and y axis.

7) Eliminate the arbitrary function f from )(z

xyfz and form the partial

differential equation.

8) Obtain partial differential equation by eliminating the arbitrary function

from ).(22 yxfz

9) Obtain the partial differential equation by eliminating the arbitrary functions f and g from ).()( itxgitxfz

10) Find the partial differential equation by eliminating the arbitrary function

from .0],[ 2

z

xxyz

11) Find the complete integral of pqqp where x

zp and

y

zq .

12) Write down the complete solution of .1 22 qpcqypxz

13) Find the singular solution of the partial differential equation

.22 qpqypxz

14) Find the complete integral of the partial differential equation .3)2()1( zqypx

15) Find the complete integral of .pqp

y

q

x

pq

z

16) Find the complete integral of .22 xqyp

17) Find the solution of .222 zqypx

18) Solve .08423

3

2

3

2

3

3

3

y

z

yx

z

yx

z

x

z

19) Solve .0)( 3223 zDDDDDD

20) Find the particular integral of ).2sin()1243( 3223 yxzDDDDDD

UNIT I PARTIAL DIFFERENTIAL EQUATIONS

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PART B

1) Form the partial differential equation by eliminating the arbitrary functions

f and g in ).2()2(33 yxgyxfz

2) Form the partial differential equation by eliminating the arbitrary functions

f and g in ).()(22 xgyyfxz

3) Form the partial differential equation by eliminating the arbitrary functions f and from ).()( zyxyfz

4) Find the singular solution of .1622 qpqypxz

5) Solve .1 22 qpqypxz

6) Find the singular integral of the partial differential equation

.22 qpqypxz

7) Solve .1 22 qpz

8) Solve ).1()1( 2 zqqp

9) Solve .4)(9 22 qzp

10) Solve .)1( qzqp

11) Solve ).)(()()( yxyxqxyzpxzy

12) Solve .2)2()( zxqyxpzy

13) Find the general solution of .32)24()43( xyqzxpyz

14) Solve )()()( yxzqxzypzyx

15) Solve ).2(2 yzxxyqpy

16) Solve .)()( 222 xyzqzxypyzx

17) Solve .)()( 22 yxzqyxzpyx

18) Solve .)2()2( xyqyzpzx

19) Find the general solution of .)( 22 qypxyxz

20) Solve ).()()( 222222 xyzqxzypzyx

21) Solve ).4sin()20( 522 yxezDDDD yx

22) Solve ).2sin(3)54( 222 yxezDDDD yx

23) Solve ).cos()( 23223 yxezDDDDDD yx

24) Solve .)30( 622 yxexyzDDDD

25) Solve .)6( 3222 yxeyxzDDDD

26) Solve .)sinh(22

22

2

2

xyyxy

z

yx

z

x

z

27) Solve .cos62

22

2

2

xyy

z

yx

z

x

z

28) Solve .sin)65( 22 xyzDDDD

29) Solve .)1222( 222 yxezDDDDDD

30) Solve .7)33( 22 xyzDDDD


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UNIT II FOURIER SERIES PART A

1) State Dirichlets conditions for a given function to expand in Fourier series. 2) Does xxf tan)( possess a Fourier expansion?

3) If )(xf is discontinuous at ax , what does its Fourier series represent at

that point?

4) Determine nb in the Fourier series expansion of )(2

1)( xxf in

20 x with period 2 .

5) If the Fourier series for the function 2;sin

0;0)(

xx

xxf is

xxxx

xf sin2

1......

7.5

6cos

5.3

4cos

3.1

2cos21)( . Deduce that

.4

2..........

7.5

1

5.3

1

3.1

1

6) Find the constant term in the Fourier series corresponding to xxf 2cos)(

expressed in the interval ).,(

7) What is the constant term 0a and the coefficient of nxcos , na in the Fourier

series expansion of 3)( xxxf in ),( ?

8) Find nb in the expansion of 2x as a Fourier series in ).,(

9) In the Fourier series expansion of

xx

xx

xf

0,2

1

0,2

1

)( in ),( , find

the value of nb , the coefficient of nxsin .

10) Find na in expanding xe as Fourier series in ).,(

11) Find the Fourier constants nb for xxsin in ).,(

12) If the Fourier series of the function 2)( xxxf in the interval x

is ,sin2

cos4

)1(3 1

2

2

n

n nxn

nxn

then find the value of the infinite series

.........3

1

2

1

1

1222

13) Find a Fourier sine series for the function 1)(xf ; x0 .

14) To which value , the half range sine series corresponding to 2)( xxf

expressed in the interval )2,0( converges at 2x ?

15) Find a0 in the expansion of | cos x | as a Fourier series in ).,(


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PART-B

1) Find the Fourier series of period 2 for the function )2,(;2

),0(;1)(xf and

hence find the sum of the series ........5

1

3

1

1

1222

.

2) Obtain the Fourier series for )2,(;2

),0(;)(

x

xxf .

3) Expand 2;0

0;sin)(

x

xxxf as a Fourier series of periodicity 2 and

hence evaluate ..........7.5

1

5.3

1

3.1

1.

4) Determine the Fourier series for the function 2)( xxf of period 2 in

20 x .

5) Obtain the Fourier series for 21)( xxxf in ),( . Deduce that

6.........

3

1

2

1

1

1 2

222.

6) Expand the function xxxf sin)( as a Fourier series in the interval

x .

7) Determine the Fourier expansion of xxf )( in the interval x .

8) Find the Fourier series for xxf cos)( in the interval ),( .

9) Expand xxxf 2)( as Fourier series in ),( .

10) Determine the Fourier series for the function xx

xxxf

0,1

0,1)( .

Hence deduce that 4

.........5

1

3

11 .

11) Find the half range sine series of xxxf cos)( in ),0( .

12) Find the half range cosine series of xxxf sin)( in ),0( .

13) Obtain the half range cosine series for xxf )( in ),0( .

14) Find the half range sine series for )()( xxxf in the interval ),0( .

15) Find the half range sine series of 2)( xxf in ),0( .

Hence find )(xf .


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APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS

PART A 1. Classify the following PDE (i) U xx = U yy (ii) U x y = U yy + xy

2. Classify the following PDE U xx + U yy = ( Ux)2 + (Uy)2

3. Write any two solutions of the transverse vibrations of a string.

4. Write down the appropriate solutions of the vibration of string

Equations. How is it chosen?

5. State any two assumptions made in the derivation of one dimension

wave equation.

6. If the ends of a string of length l are fixed and the mid point of the String is drawn aside through a height h and the string is released

from rest, write the initial conditions.

7. When a vibrating string has an initial velocity, its initial conditions

Are.?

8. When a vibrating string has an initial velocity zero, its initial

conditions Are.?

9. State one dimensional heat equation with the initial and boundary conditions.

10. State the laws assumed to derive the one dimensional heat equation. 11. Define steady state temperature distributions. 12. When the ends of a rod length 20cm are maintained at the

temperature 10oc and 20oc respectively until steady state is prevail.

Determine the steady state temperature of the rod.

13. In steady state conditions derive the solution of one dimensional heat flow equation.

14. State Fourier law of heat conduction. 15. Write down the appropriate solutions of the one dimensional heat

Equations. How is it chosen?

16.What is the basic difference between the solutions of one dimensional

wave equation and heat equation.

17. Write down the partial differential equation that represents variable Heat flow in two dimensions? Deduce the equations of steady state

heat flow in two dimensions.


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18. Write down the appropriate solutions of the two dimensional heat

Equations.

19. In two dimensional heat flow, what is the temperature along the Normal to the xy- plane?

20. If a square plate has its faces and the edge y = 0 insulated, its edges

x = 0 and x = n are kept at zero temperature and its fourth edge is

kept at temperature u, then what are the boundary conditions for this

problem?

PART B

21. A tightly stretched string with fixed end points x = 0 and x = l is initially in a position given by y = y0 sin3( x / l ). If it is released from

rest from this position, find the displacement y( x, t).

22. A tightly stretched string of length l has its ends fastened at x = 0 , x = l. The mid-point of the string is then taken to height h and then

released from rest in that position. Find the lateral displacement of a

point of the string at time t from the instant of release.

23. A tightly stretched string with fixed end points x = 0 and x = l. At time t = 0, the string is given a shape defined by F(x) = x ( l - x ), where

is constant, and then released . Find the displacement of any point x of

the string at any time t >0.

24. The points of trisection of a string are pulled aside through the same distance on opposite sides of the position of equilibrium and the string

is released from rest. Derive an expression for the displacement of the

string at subsequent time and show that the mid-point of the string

always remains at rest.

25. A tightly stretched string of length l with fixed ends is initially in equilibrium position. It is set vibrating by giving each point a velocity

v0 sin3( x / l ). Find the displacement y(x,t).

26. A tightly stretched string with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. . It is set vibrating by giving

each point a velocity x ( l - x ), find the displacement of the string at

any distance x from one end at any time t.


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27. A taut string of length 20cms.fastened at both ends is displaced from its

position of equilibrium , by imparting to each of its points an initial velocity

given by: v = x in 0 < x < 10 and, x being the

20 x in 10 < x

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35. Find the steady temperature distribution at points in a rectangular plate With insulated faces the edges of the plate being the lines x = 0 , x = a ,

y = 0 and y = b. When three of the edges are kept at temperature zero

and the fourth at fixed temperature a0c.

36. Solve the BVP Uxx + Uyy = 0 , 0 < x, y < with u( 0 ,y ) = u( , y) = u( x , ) = 0 and u(x,0) = sin3x.

37. A rectangular plate is bounded by the lines x = 0 , y = 0 , x = a , y = b . Its Surfaces are insulated. The temperature along x = 0 and y = 0 are kept

at 00 c and the others at 1000 c . Find the steady state temperature at any

point of the plate.

38. A long rectangular plate has its surfaces insulated and the two long sides as well as one of the short sides are maintained at 00 c . Find an

expression for the steady state temperature u(x,y) if the short side y = 0

is cm long and is kept at uo0c.

39. An infinitely long rectangular plate with insulated surface is 10cm wide. The two long edges and one short edge are kept at zero temperature

while the other short edge x = 0 is kept at temperature given by

U = 20y for 0 < y < 5

20 ( 10 y ) for 5 < y < 10 .Find the steady state temperature

distribution in the plate.

40. An infinitely long plane uniform plate is bounded by two parallel edges

and an end at right angle to them. The breadth of this edge x =0 is , this

end is maintained at temperature as u = k (y y2) at all points while the

other edges are at zero temperature. Determine the temperature u(x,y)

at any point of the plate in the steady state if u satisfies Laplace equation.


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Unit IV

FOURIER TRANSFORMS

PART A

1. State the Fourier integral theorem.

2. Define Fourier transform pair.

3. State the convolution theorem on Fourier transform.

4. Write the Parsevals identity for Fourier transform.

5. State Modulation theorem on Fourier transform.

6. Prove that if F(s) is the Fourier transform of f(x), then F{ f(x a) } = eisa F(s).

7. State and prove change of scale property of Fourier transform.

8. Find the F.T of f(x) defined by f(x) = 1 if a a >0

10. Find the F.T of f(x) defined by f(x) = eimx if a

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PART B

1. A function f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Using Fourier

integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in 0 < x< .

2. A function f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Using Fourier

cosine integral representation of f(x). Hence evaluate [sin s/ s] cos sx ds in

0 < x < .

3. Find the F.T of f(x) is defined as f(x) = 1 if |x| < 1

0 , otherwise . Hence evaluate

(i) [sin / ] d (ii) [sin

2 /

2]

d in (0 , ).

4. Find the F.T of f(x) is defined as f(x) = a -|x| if |x| < a


(i) [sin / ] 2

d in (0 , ). (ii) [sin / ] 4 d in (0 , ).

5. Find the F.T of f(x) is defined as f(x) = 1 x2 if |x| < 1


(i) [sin t t cos t/ t 3] dt in (0 , ). (ii) [x cos x - sin x / x 3] cos (x /2)dx in (0 , ).

6. Find the F.T of f(x) is defined as f(x) = e-a2x2

,a >0. Hence S.T e-x2 / 2

is self reciprocal

under F.T.

7. Find the F.T of e-|x|

and hence find the F.T of e-|x|

cos 2x.

8. Obtain the F.S.T of f(x) = x if 0

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reciprocal under both F.S.T and F.C.T.

12. Find the F.S.T of e-ax

/ x . Hence find F.S.T of 1 / x.

13. Evaluate [dx / (a2 + x

2 ) (b

2 + x

2) ]

dx in (0 , ).

14. Find F c {f (x)}.

15. Solve the integral equation [f(x) cos x] dx in (0 , ) and also [cos x / ( 1 +

2)]

d

in (0 , ).


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Z TRANSFORM

PART A

1. Define Z transform of sequence Un.

2. Find Z [ an ]

3. Find Z [ n ]

4. Find Z [ c ], where c is any constant.

5. Find Z [ cos n ]

6. Find Z [ sin n ]

7. State Damping rule.

8. Define Z transform of f(t)

9. Find Z [ eat ]

10. Find Z [ t ]

11. Find Z [ eat f(t) ]

12. State second shifting property of Z transform.

13. State convolution theorem of Z transform.

14. State final value theorem.

15. State initial value theorem.

16. Find Z [ 1 / n ]

17. Find Z [ 1 / n! ]

18. Find Z [ 1 / n ( n + 1) ]

19. Find Z [ 2 n2 + 3 n + 7 ]

20. Find Z [ np ]

PART B

21. Find Z [ an cos n ] and Z [an sin n ]

22. Find Z [an n2 ]. 23. Find Z [ cos n/2 ] and Z [ sin n/2 ]

24. Find the Z transforms of the following (i) ean (ii) n ean

25. Find the Z transform of (i) cosh n (ii) an cosh n 26. Find Z [ cos ( n/2 + /4 ) ]


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27. Find the Z transform of (i) ncp (ii) n+p cp 28. Find the Z transform of unit impulse sequence and unit step sequence.

29. Find the Z transform of (i) sinh n (ii) an sinh n

30. Find Z [ et sin2t ] and Z [ e-2t sin3t ].

31. Find the inverse Z transform of z / ( z + 1 )2 by division method.

32. Find the inverse Z transform of { 2 z2 + 3z } / ( z + 2) ( z 4 ) by partial

fractions method.

33. Find the inverse Z transform of ( z3 20 z ) / ( z 2 ) 3 ( z 4 ) by partial fraction method.

34. Find the inverse Z transform of 10 z / ( z-1) ( z-2) by inversion integral method.

35. Find the inverse Z transform of 2z / ( z -1 ) ( z - i ) ( z + I ) by inversion integral method.

36. Using convolution theorem , evaluate the inverse Z transform of z2/ ( z -a ) ( z - b )

37. Using convolution theorem , evaluate the inverse Z transform of z2/ ( z a) 2

38. Show that ( 1/ n! ) * (1/ n! ) = 2n / n! 39. Solve yn+2 + 6 y n+1 + 9yn = 2n with y0 = y1 = 0, using Z transform. 40. Solve yn+2 - 2 y n+1 + yn = 3n + 5.


UNIT-I (PDE).pdfUNIT-II-FOURIER SERIES.pdfUNIT-III-APP OF PDE.pdfUNIT-IV-FOURIER TRANS.pdfUNIT-V-Z-TRANS.pdf