ANAND INSTITUTE OF HIGHER TECHNOLOGY

TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Unit-I Fourier Series

Part-A (Question and Answers)

1. State Dirichlet’s condition (May/June 2013,2009)

Sol: (i) f(x) is bounded, single valued f(x) has atmost a finite no. of maxima and minima f(x) has atmost a finite no. of discontinuities

2. Write the formula for finding Euler’s constant of a Fourier series in (0,2π) (A.U 2009)

Sol:

where

3. Sum the Fourier series for at x = 0 and x =1.(A.U N/D 2010)

Sol: Here x =0 is discontinuous point

Here x = 1 is discontinuous point

4. What is the constant term ao and the coefficient of cosnx, an in the Fourier series expansion of f(x)=x-x3 in (-π, π).(A.U N/D 2010)

Sol:

f(x) is odd functionao = an = 0

5. State Parseval’s Identity for the full range expansion of f(x) as Fourier series in (0, 2 l ).(A.U 2008)

1

Sol:

6. Find a Fourier sine series for the function f(x) = 1, 0 < x < π (A.U 2007 nov 2013)

Sol:

7. If the Fourier series for is

Prove that (A.U 2009)

Sol:

Put in ( 1 )

Here is continuous point .

(1)

8. Find bn in the expansion of x2 as a Fourier series in (-π, π).(A.U 2005,2008) Sol: Here is even function in (-π, π)

2

9. If f(x) is an odd function defined in (-l , l ) what are the values of a0 and an.?(A.U A/M 2010) Sol: Here f(x) is an odd function in (-l , l )

10. What do you mean by Harmonic Analysis? (May/June 2013,2010)

Sol: The process of finding the Fourier series of the function f(x) which is given in terms of numerical value.

11. Find an in expanding e -x as Fourier series in (-π, π) Sol:

12. State Parseval’s Identity for the Half range cosine series expansion of f(x) in (0,1).A.U 2007)

Sol:

13. If for 0<x<l the function f(x) has the expansion .Show that

(A.U 2007) Sol:

WKT

Given ---- (1)

Multiply f(x) in the above eqn.(1) and integrate over 0 to l,

3

14. Find the value of an for f(x)=k in (0,10) in cosine series expansion.(A.U 2006) Sol:

15. Define Root Mean Square value for f(x) in (a,b) (Nov/Dec 2012)

Sol:

16. Find a0 in expanding e -x as Fourier series in (o, 2π) Sol:

PART – B Questions:

1. Obtain the Fourier series of period 2l for f(x) where f(x) = . Hence find sum of

. And (Nov 2004, 2003 May 2001, Nov 2007,2012)

2. Expand f(x) = as a Fourier of the periodicity 2 and hence evaluate

. (May 2007)

3. Determine the Fourier series for the function f(x) = of period 2 in 0 < x < 2 .(May 2007)4. Find the Half range cosine series for the function f(x) = x( -x) in 0 < x < , Deduce that

4

.(May 2007,2013)

5. Find the complex form of Fourier series for the function f(x) = .(May 2007)6. Determine the Fourier series for the function f(x) = Deduce that

. (May 2007,2013)

7. Find the Fourier series for f(x) = .( April 2005,Dec 2005,May 2006,2010)8. Find the half range sine series f(x) = x( -x) in (0, ) and Deduce that

(May 2006)

9. Find the Fourier series up to second harmonic for the following data X: 0 1 2 3 4 5 Y: 9 18 24 28 26 20. (May 2006,2010) 10. Expand f(x) = - x, as Fourier series in (- , ). (May 2006) 11. Find half range cosine series given f(x) = (May 2006) 12. Find the Fourier series of period 2 as for as second harmonic given 30 60 90 120 150 180 210 240 270 300 330 360 3.01 3.69 4.15 3.69 2.2 0.83 0.51 0.88 1.09 1.19 1.64 2.34 (May 2006,2012)

13. Find the Fourier series expansion of period l for the function f(x) =

Hence deduce the sum of the series. ( Nov 2004, 2006,Apr 2005, NOV 2013)

14. Find the half range cosine series of f(x) = ( - ) in the interval (0, ) Hence find the sum of

Series . (Nov 2006)

15. Find the Fourier series of f(x) = and hence find the sum of the series

. ( Nov 2005, 2006, May 2006, NOV 2013)

16. Find the Fourier series for f(x) = in (- , ) and hence find .

Hence Deduce that and (Nov 2005,May 2001)

17. Obtain a Fourier expansion for in (- <x< ). (Nov 2005)

18. Obtain the cosine series for f(x) = x, in 0<x< and deduce that .

5

(Nov 2005, Nov 2008)

19. Find the Fourier series for the function f(x) = Deduce that

. (Nov 2005)

20. Find the first fundamental harmonic of the Fourier series of f(x) given by the following table X: 0 1 2 3 4 5

Y: 9 18 24 28 26 20. (Nov 2005) 21. Determine the Fourier expansion of f(x) = x in the interval - <x< (Apr 2004)

22. Find the half range cosine series for xsinx in (0, ) (Apr 2004)

23. Obtain the Fourier series for the function f(x) == (Apr 2004)

24. Find the Fourier series of f(x) = (Nov 2004)

25. Find a0 , a1 , a2 , b1 , b2 , b3 given

X : 0 2

. Y : 1.0 1.4 1.9 1.7 1.5 1.2 1.0 (Nov 2004,Apr 2008,NOV 2013) 26. Obtain Fourier series for f(x) of period 2l and defined as follows

f(x) = Hence deduce that and

(Nov 2004, 2003 May 2001)

27. Obtain Fourier series for f(x) = in (- , ), deduce that (A.U 2003)

28. Obtain Fourier series for f(x) of period 2L and defined as follows

f(x) = Hence deduce that

29. Express f(x) = as a Fourier series with period 2 to be valid in the interval (0,2 )

Hence deduce the value of the series . (Apr 2001)

30. Find the Fourier series for f(x) = and Hence deduce the sum of the series

. (May 2004)

6

31. Find the Fourier series of the function f(x) = .and Hence Evaluate

. (Apr 2000).

32. Determine the Fourier series for the function f(x) = of period 2 in 0 < x < 2 l.(A.U 2007)

33. Find the half range sine series expansion of in (0, ) and deduce the sum of the series

. (Dec 2007)

34. Find upto the first two harmonics in the fourier series of y = f(x) in (0,360) given in the following tabular value.

X Y 2 2.1 3 3.2 2.5 2.2 2 (A.U 2007,2012)

35. Expand in Fourier series of f(x) = x sinx for 0< x < 2 and deduce the

result ( A.U Apr 2008)

36. Obtain half range sine series for f(x) = x in 0< x < l and deduce the series = . (A.U 2008)

37. Find the Fourier series of periodicity 3 for in 0< x < 3. (A.U Apr 2008)

38. Find the complex form of Fourier series for the function f(x) = , (A.U May 2008,2012).

39.Find the fourier series expansion of f(x) = .(NOV 2013)

40.Find the half range series of f(x)=lx- in (0,l) (NOV 2013)

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Unit-II Fourier Transform

Part-A (Questions and Answers)

1. State Fourier Integral theorem (A.U 2011,2010)

Sol:

7

2. State Fourier Transform Pair (A.U 2011)

Sol:

3. State Fourier Sine Transform Pair (A.U 2011)

Sol:

4. State Fourier Cosine Transform Pair (A.U 2011)

Sol:

5. Find Fourier Sine Transform of (A.U M//J 2013,2012)

Sol: W.K.T.

=

=

=

6. Find Fourier Cosine Transform of (A.U 2009)

Sol: W.K.T.

8

=

=

=

7. Solve the Integral Equation (A.U 2005)

Sol: W.K.T.

= =

=

= =

8. Define Self Reciprocal (A.U 2011,NOV 2013)

Sol: If the Fourier Transform of f(x) is f(s) then f(x) is said to be self reciprocal function.

9. State Parsevel’s Identity (A.U 2010,2011)

Sol:

10. State Convolution theorem (A.U 2010)

Sol:

11. Prove that where (A.U 2010)

9

Sol:

=

= F(s + a)

12. Prove that where (A.U M/J 2013,2011)

Sol:

Let u = a x then x = u / a and hence dx =

=

13. Find Fourier Sine Transform of (A.U 2011)

Sol:

=

W.K.T

Let

14. Find Fourier Transform of

(A.U 2009)

10

Sol:

15. Prove that where (A.U 2009)

Sol:

=

=

=

=

=

16. Prove that (A.U 2011)

Sol:

17. Prove that where (A.U 2010)

11

Sol:

=

= F(s -a)

*****

PART –B Questions

1. Find the Fourier Transform of . Hence deduce that .

(Apr 2001, May 2005,2006,2007, Nov 2005,Dec 2012)

2. Evaluate using Fourier Transform. ( Apr 2001,2006, Nov 2005, Dec 2010)

3. Find the Fourier transform of Hence prove that (i) )

. ( Apr 2000, 2001, May 2003, 2005, 2006,2007, Nov 2004,June 2010 ,NOV 2013)

) 4. Find the Fourier Cosine Transform of ? ( Nov 2005, 2006)

5. Find the Fourier Transform of if . Deduce that (i) and

(ii) . ( Nov 2002, 2005)

6. Prove that is the self reciprocal under Fourier Cosine Transform, Deduce that is the

self reciprocal under Fourier Sine Transform. ( Nov 2005)

7. Find the Fourier Sine Transform of , where a>0. ( May 2005, Nov 2006)

8. Find the Fourier Cosine Transform of . ( Nov 2004,Dec 2009,June 2011)

9. Show that Fourier Transform of is . Hence deduce

that . Using Parseval’s identity show that .

( Nov 2003, may 2004,Dec 2010,Dec 2012)10. Find the Fourier Sine Transform of ( May 2004,Dec 2009,Nov

2010,Nov 2011)

11. Find the Fourier Transform of f(x) given by and hence evaluate

and ( Apr 2003, Nov 2003)

12. Find the Fourier Sine Transform and Fourier Cosine Transform of and hence find the Fourier

12

Sine Transform of and Fourier Cosine Transform of .( Apr 2003, Nov 2003,Dec 2012)

13. Find the Fourier Transform of if x>0. Deduce the if a>0. ( May 2003,NOV2013

)

14. Find the Fourier Sine Transform of .( May 2003)

15. Derive the Parseval’s Identity for Fourier Transform. ( May 2003,June 2012)16. Find the Fourier Sine Transform of. ( Nov 2002, May 2007,May 2010)

17. Find the Fourier Cosine Transform of . Deduce that and .

(Nov 2002)18. State and prove Convolution theorem for Fourier Transform. ( Nov 2002,June 2012,May 2010,Apr 2009)

19. Find the Fourier Transform of if a>0. DeduSce that . ( May 2007,2011)

20 Prove that is self reciprocal under Fourier Cosine Transform. ( May 2007,Dec 2012,June 2013)

21. Find the Fourier Sine Transform of. , a>0. and hence deduce the inversion formula. ( May 2007,Dec 2012)

22. Solve for from the integral equation .( May 2007)

23. Find the Fourier Sine Transform and Fourier Cosine Transform of . (May 2006 ,June 2012)24. Find the Fourier Sine Transform of. , a>0 hence find .(May 2006 ,Dec 2009)

25. Using Parseval’s Identity for Fourier Cosine Transform of , evaluate . (May 2006 )

26. Find the Fourier Transform of . Hence prove is self reciprocal. (May 2006 ,Dec 2011)

27. Use Transform method to evaluate .( Nov 2006,June 2010)

28. If is the Fourier Transform of , Find the Fourier Transform of and . ( May 2004)

29. Verify Parseval’s theorem of Fourier Transform for the function ( May 2004)30. Find the Fourier Cosine Transform of ( Apr 2001,June 2012)

31. Find the function whose fourier sine transform is (a>0) NOV 2013

13

Unit – III PDE

Part – A Questions and Answers

1. Form the PDE by eliminating arbitrary constants a and b from z = (x+a)(y+b).(A.U 2010)

Ans: Given z = (x+a)(y+b)…..(1) Partially Differentiating eqn (1)with respect to x & y we get

….(2)

……(3)

Eliminating a & b between eqns (1) ,(2) & (3) (ie) Substituting (2)and (3) in (1) we get z = pq.

2. Find the PDE of all planes having equal intercepts from the x and y axis.(A.U 2009)

Ans:

Eqn. of such plans is …(1)

Partially differentiating (1) w.r.t x & y we get

From (2) & (3) we get p = q. which is the required PDE.

3. Eliminate ‘f’ from z = f(y/x) (A.U 2009,2012)

Ans: Given z = f(y/x)……(1) Partially differentiating (1) w.r.t x & y we get

14

From (2) & (3) we get px +qy = 0.

4. Form the p.d.e by eliminating f from z = f(x+y).(A.U 2010)

Ans: Given z = f(x+y)……(1)

Differntiating (1) p.w.r.t. x &y we get

&

5. Solve (A.U 2007,2011)

Ans:

Given Integrating

6. Write the complete solution of p + q = x + y. (A.U 2008)

Ans: Let p + q = x + y = k p – x = k, y – q = k., ie p = k + x, q = y – k

7. Find the complete integral of (1) ( A.U 2010)

Ans: The given equation is Clairaut’s type

Put p = a & q = b in equation (1)Complete integral is

8. What is the C.F of (A.U 2007)

Ans: Replace D by m and D/ by 1

A.E is m2 – 4m + 4 = 0.(m – 2)2 = 0.C.F = f1(y + 2x) + xf2(y + 2x).

9. Solve (A.U 2009)15

Ans: A.E is m3 – 3m + 2 = 0.

m = 1, 1, – 2. z = f1(y + x) +xf2(y + x) +f3(y – 2x).

10. Find the particular integral of (A.U 2008)

Ans:

P.I =

Replace D2 by – 1, DD/ by – 5, D/2 by – 25

P.I =

11. Find the particular integral of (A.U 2011)

Ans:

P.I = Replace D by 1, D/ by 0.

P.I = ex.

12. Write the solution of pq = 2 ……(1) (A.U 2006)

Ans: Let z = ax +by + c………… (2) be the solution

Put P = a. q = b. in equation (1) => ab = 2 => b = 2/a Substitute b = 2/a in equation (2) we get Z = ax + (2y)/a + c. This is the required solution.

13. Find the general solution in terms of arbitrary functions for the PDE 2p + 3q = 1.(A.U 2009)

Ans:

Subsidiary equation is

Consider integrating 3x – 2y = c1

Consider integrating y – 3z = c2

The Solution is f(3x – 2y, y – 3z) = 0.

14. Explain the method of solving Lagrange’s Linear equation (A.U 2010)

Ans.: Lagrange’s linear P.D.E is of the form Pp + Qq = R where P, Q, R are functions of x,y,z,

To solve this, we first form the subsidiary simultaneous equations

If u = a and v = b Are two independent integrals of these simultaneous equations, Then the solution of P.D.E. is Ф (u, v) = 0 where Ф is an arbitrary function.

16

It can also be put as u = f (v), f being arbitrary.

There are two methods of solving the simultaneous equations (1) Method of grouping

(2) Multiplier Method.

15. Solve (A.U 2009)

Ans: A.E is m2 + 2m = 0. m (m+2) = 0 => m =0, m= – 2

The Solution is z = f1(y) +f2(y – 2x)

PART – B Questions

1. Solve (May 2007)2. Solve (May 2007)3. Solve (May 2006,2009)4. Solve (May 2006,2010)5. solve (May 2006)6. solve (Dec 2006)7. solve (may 2005,2012)8. solve (Dec 2006)9. solve (May 2005)10. solve (Dec 2005)11. solve (Nov 2005)12.Solve (Nov2005)13.Solve (May 2004, Apr 2000,2008,2013)14.Solve (Dec 2004)15.Solve (Dec 2003,2011)16.Solve (Dec 2003)17.Solve (Apr2000)18.Solve 19.Solve (Apr2001)

20.Solve (Apr2004,2010)

21.Solve (May2007,2008)22.Solve (May 2007)23.Solve (May2007)24.Solve (May2007,2008,2009,2012)25.Solve (May2006)26.Solve (May2006)

17

27.Solve 28.Solve (A.U 2008)29.Solve (Nov2006)30.Solve (may 2006)31. Solve (may 2005,2010,2012)

32. solve (Dec 2005)33. Solve (Dec 2005,2011)34. solve (Dec 2005)35. Solve (Dec 2005,2011,2013, NOV 2013)

36. Solve (Dec 2004,2008,2013)

37. Solve (may 2004,2008,2011)38. Solve (Dec 2004)39. Solve (Dec 2002,2011)40. Solve (Apr 2003)41. Form the Partial Differential Equation from by eliminating the arbitrary constants ‘a’ and ‘b’ from the expression (May 2007)42.Find the Singular integral of (Nov2006)43.Find the Singular integral of (Nov2003,2010)44.Find the Singular solution of (Nov2002)45. Form the PDE by eliminating f & from (Dec 2004)46. Find the C.I of p +q = x + y47. Form the PDE by eliminating the arbitrary function from the relation

(April 2003,2009)48. Form the PDE by eliminating the arbitrary function from the relation

(Nov 2002,2011)49. Form the PDE by eliminating the arbitrary function from the relation

(April 2003,2007,NOV 2013)50. Form the PDE by eliminating the arbitrary function from the relation

(Nov 2003)51. Form the PDE by eliminating the arbitrary function from the relation

(April 2004) 52. Solve NOV 201353.Solve NOV 2013

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UNIT – IV APPLICATIONS OF PDE18

1. Write all the solution of the wave equation (A.U 2010,2011) Sol.: The wave equation is . The various possible solution of this equation is

y(x,t) =

=

= .

The pde of a vibrating string is what is ?

Solution: =T/m = Tension/mass per unit length of string.

2. State the assumptions made in the derivation of one dimensional wave equation.(A.U 2011) Sol.: The wave equation is (i) The motion takes place entirely in one plane.ie xy plane. (ii) We consider only transverse vibrations. The horizontal displacement of the particles of the string is negligible. (iii) The tension T is constant at all times and at all points of the deflected string. (iv) T is considered to be so large compared with the weight of the string and hence the force of gravity is negligible. (v)The effect of friction is negligible. (vi) The string is perfectly flexible.

3. Give reasons for choosing y = (Acospx+Bsinpx)(Ccospat+Dsinpat) as a suitable solution of the pde of the vibrating string.

Sol.: Since the vibrating of the string is a periodic motion with respect to time .we must get a solution for y(x,t) in which trigonometric terms of t are present.

4. Write down the boundary conditions for the following boundary value problem. “if a string of length ‘ℓ’ is initially at rest in its equilibrium position & each of its points is given the

velocity determine the displacement functions.

Sol.: wave equation boundary conditions are y(0,t)=0 y(ℓ,t)=0}, t>0 y(x,0)=0

5. Write all the solution of one dimensional heat equation. (A.U M/J 2013) Sol.: u(x,t) =

u(x.t )=

u(x,t)= .

6. The partial differential equation of one dimensional heat equation is what is ?(A.U 2011) Sol.: is called the diffusivity of the material of the body through which heat flows.

19

If ρ be the density, c the specific heat & k thermal conductivity of the material. We have the relation = k/ρc.

7. What is meant by steady state condition in heat flow? (A.U 2001) Sol.: steady state condition in heat flow means that the temperature at any point in the body does not vary with time. It is independent of ‘t’ the time.

8. In steady state conditions derive the solution of one dimensional heat flow? (A.U 2011,nov 2013) Sol.: The pde of unsteady one dimensional heat flow is In steady state condition the temperature u depends only on x & not time t. Hence . Eqn (1) reduces to solving 2. The general solution of 2 is u = ax + b where a, b are arbitrary.

9. Find the steady state temperature of a rod of length l whose ends are kept at 30 and 40 . Sol.: The steady state equation of one dimensional heat flow is Solving we get u = a x+ b . The boundary conditions u = 30 when x = 0, u = 40 when x = ℓ 30 = a(0) + b, => b = 30 40 = a (ℓ) + 30, => a ℓ = 10, => a = 10/ℓ Sub in 2 we get u = 10x/ ℓ + 30.

10. Explain the term “ Thermally insulated ends ‘ Sol.: If an end of a heat conducting body is thermally insulated. It means that no heat passes through that section Mathematically the temperature gradient is zero at that point .

11. Express the boundary conditions in respect of insulated ends of a bar of length ‘a’ and also the temperature distribution f(x). (A.U 2011)

Sol.: The boundary conditions are

For all values of t

u(x,0) = f(x) for 0 < x < a.

12. What are the assumptions made while deriving one dimensional heat equation? Sol.: 1. Heat flows from a higher to lower temperature. 2. The quantity of heat required to produce a given temperature change in a body is proportional to the mass of the body and the temperature change .The constant of proportionality is known as the specific heat(c) of the material. 3. The rate at which heat flows across any area is jointly proportional to the area and to the temperature gradient normal to the area. Ie The rate of change of temperature w.r.t the distance normal to the area . The constant of proportionality is known as the thermal conductivity (k).This is known as Fourier law of heat conduction.

20

13. What is steady state heat equation in two dimensions in Cartesian form? Sol.: The required equation is

14. Write the different solutions of Laplace equation in Cartesian coordinates? (A.U 2008) Sol.:

15. For write a solution which is periodic in y?

Sol.: 16. Write the general solution y(x,t) of vibrating motion of a string of length ‘ℓ’ with fixed end point and zero initial

velocity. Sol.:

y(x,t) =

17. A rectangular plate is bounded by the lines x = 0, y = 0, x = a and y = b. Its surfaces are insulated and the temperature along the adjacent sides x = a; y = b are kept at 100 & the temperature along the two sides x = 0 & y = 0 are kept at 0.write the boundary conditions.

Sol.: The boundary conditions are u(0,y) = 0 u(a,y) = 100}, 0 u(x,0) = 0 u(x,b) = 100}, 0

18. classify the pde (i)

(ii)(

Sol.: (i) Here A = 1; B = 0; C = x – 4AC = 0 – 4x = – 4x

If x is negative (x < 0) then ie (i) is Hyperbolic

If x > 0 then is elliptic

If x = 0 then is parabolic

(ii) Here A = ; B = – 2xy; C =

Now (ii) is parabolic. 19. Classify the following (i) (ii) Sol.: (i). A = x; B = 0; C = y = – 4xy < 0

is elliptic

(ii). A = 1; B = – 2; C = 0

21

is Hyperbolic. 20. Classify the following pde?

Sol.: Here A = ; B = 0; c = 1

– 4AC = 0 – 4 = – 4 < 0

is elliptic

21. An insulated rod of length I cm has its ends A and B maintained at and 80⁰crespectively. Find the steady state

solution of the rod.(nov 2013)Soln. U=ax+b X=0 => U(0)=b, b=0 X=1 => u(l)=al+b 80=al a=80/l

Part-B Questions

1. A tightly stretched flexible string has its ends fixed at x = 0 and x = l. At the time t = 0, the String is given a shape defined by where k is a constant and then Released from rest. Find the displacement at any point ‘x’ of the string at any time .

(May2005 NOV 2013)

2. A uniform string is stretched and fastened to two points ‘l’ apart. Motion is started by displacing the string into the form of the curve and then releasing it from this position at time t = 0. Find the displacement of the point of the string at a distance x from one end at time t.

(Nov 2003, 2007, May 2007, 2008,2009,2013)

3. An elastic string of length 2l fixed at both ends is disturbed from its position at equilibrium Position by imparting to each point an initial velocity of magnitude . Find the displacement function . (May 2006, Nov 2007)

4. A tightly stretched string with fixed end points is initially in a position

equation by .It is released from rest from this position. Find the displacement

at any time‘t’ (Nov 2004)

5. A string is tightly stretched and its ends are fastened at two points at .The midpoint of the string is displaced transversely through a small distance ‘b’ and the string is released from rest in that position. Find an expression for the transverse displacement of the string at any time during the subsequent motion? (Nov 2002, 2005, May 2005, April 2001,2010)

6. A tightly stretched string of length 2l is fixed at both ends the midpoint of the string is displaced by a distance ’b’ transversely and the string is released form rest in this position. Find the

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displacement of any point of the string at any subsequent time. (Nov 2006, 2005, 2002, May 2005,2012)

7. A taut string of length l has its ends fixed. The point where is drawn aside a

small distances h, the displacement y(x, t) satisfies . Determine at any time‘t’?

(May 2006)

8. A stretched string with fixed end points is initially at rest is in equilibrium position. if it is set vibrating giving each point a velocity then show that

(Nov 2003, April 2001)

9. If a string of length l is initially at rest in its equilibrium position and each of its points is given

a velocity V such that V S.T the displacement any time t is given by

(May 2007)

10. A string is stretched between two fixed points at a distance 2 l apart and the points of the

string are given initial velocities V and V = x being the distance from an

end point. Find the displacement of the string at any time? (May 2004, 2008, Dec 2008)

11. A tightly stretched string with fixed end points is initially at rest in its equilibrium

Position. If it is set to vibrate by giving each point a velocity , find the displacement

Of the string at any subsequent time. (Dec 2008,2012)

12. A rod of length l has its end A and B kept at c and c respectively. Until steady state conditions prevail. If the temperature at B is reduced suddenly to c and at the same time the temperature at A raised to c . find the temperature at a distance x from A and at time t.

(May 2003).

13. A metal bar 10cm long with insulated sides, has its ends A and B kept at c and c respectively. until steady state conditions prevail. The temperature at A is suddenly raised to c and that the same same time that at B is lowered to c. find the subsequent temperature at any point of the bar at any time. Hence prove the temperature at the mid point of the rod remains for all time , regardless of the material of the rod. (May 2001, 2003, Nov 2005)

14. The ends A and B of a rod l c.m. long have their temperatures kept at c and c., until steady state

23

conditions prevail. The temperature of the end B is suddenly reduced to c and that of a is increased to c. find the temperature distribution on the rod after time t. (May 2007,2012)

15. A rod of length 20cm has its ends A and B kept at temperature c and c respectively until steady state conditions prevail. It the temperature at each end is then suddenly reduced to c and maintained c. find the temperature distribution at a distance from A at time ‘t’.(Nov 2005)

16. A rod of length 30 cm has its ends A and B kept at respectively until steady state conditions prevail. If the temperature of A is suddenly raised to while that the other end B is reduced to , find the temperature distribution at any point in the rod. (Dec 2008,2012)

17. 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 is kept at temperature given by

, Find the steady state temperature distribution in the plate?

(May 2006, Nov 2004, 2005,2013)

18. A rectangular plate with insulated surface is 8cm wide and 10cm long compared to its width that it may be considered infinite in the length without introducing an appreciable error. It the

temperature along one short-edge is given by . While the

two long edges and as well as the other short edges are kept at c. find the steady state temp. for . (Nov 2003)

19. A rectangular plate with insulated surfaces is ‘a’ an wide and so long compared to its width that it may be considered infinite in length, without introducing an appreciable error. It the two long edges and the short edge at infinity are kept at temperature c, while

the other short edge is kept at temperature , find the steady state temperature

at any point of the plate. (May 2007)

20. An infinitely long plate in the form of an area is enclosed between the line of. the temperature is zero along the edges and the edge at infinity. If the edge is kept at temperature find the steady state temperature distribution in the plate. (May 2006)

21. An infinitely long uniform plate is bounded by two parallel edges and an end at right angle to them. The breadth of this edge is , this end is maintained at temperature as at all points while the other edges are at zero temperature. Find the temperature at any point of the plate in the steady state. (Nov 2006).

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22. A rectangular plate with insulate surface is 10cm wide and so long compared to it’s with that it may be considered infinite in length without introducing appreciable error. The temperature at short edge y = 0 is given by

and all the other three edges are kept at c. Find the steady state

temperature at any point in the plate. (May 2005, 2008)

23. A rectangular plate with insulated surface is 10cm wide so long compared to its width that it may be considered infinite in the length without introducing an appreciable error. It the

temperature along one short-edge is given by . While the

two long edges and as well as the other short edges are kept at c. find the steady state temp. for . (Nov 2003,2012)

24. A rectangular plate with insulate surface is 10cm wide and so long compared to it’s width that it may be considered infinite in length without introducing appreciable error. The temperature at short edge y = 0 is given by

as well as the other short edges are kept at c. Find the temperature

at any point of the plate in the steady state. (May 2009)

25. Find the solution to the equation . That satisfies the condition u(0,t)=0, u(l,t)=0, for t>0 and

u(x,0) (nov 2013)

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UNIT-V Z – Transforms

Part – A

1. Define Z- transform. (A.U 2013) Sol.: If {x(n)} is a causal sequence if x(n) = 0 for n< 0, then Z transform is called one sided or unilateral

Z transform of {x(n)} and is defined as Z{x(n)} =

2. Prove that Z-transform is linear. (Or) Prove that Sol.:

3. Find Z{an}. (A.U 2008) Sol.:

4. Find Z{eat} (A.U 2010) Sol.:

5. Find Sol.:

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6. Find Z-transform of ‘n’ Sol.:

7. Find Z{n2} (A.U 2011) Sol.:

8. Find Z transform of 1/n. Sol.:

9. Find Z transforms of (A.U 2009)

Sol.:

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10. Prove that (A.U 2011) Sol.:

11. State and prove initial value theorem. (A.U 2010) Sol.:

12. If Sol.:

13. Find Z{nan}28

Sol.:

14. Find Z{n(n – 1)} (A.U 2010) Sol.:

15. Find Z-transform of ‘t’ Sol.:

16. Prove That (A.U 2008) Sol.:

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17. Find (A.U 2007) Sol.:

18. Find ( A.U 2011) Sol.:

19. Find (A.U 2010) Sol.:

20. Find Sol.:

21Find Z transforms of (A.U nov 2013)

Sol.:

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PART – B Questions

1. Find using the partial fractions. (A.U 2010)

2. Solve the difference equations where y(0) =1, Y(1)=0 (A.U 2008)

3. Prove that ( A.U 2007)

4. State and prove second shifting theorem in Z- transform (A.U 2013)

5. Using convolution theorem evaluate inverse Z- transform of (A.U 2009,nov 2013)

6. Solve the difference equation given that Y(0)=3 and Y(1)=-2.7. Solve the equation given that Yo=Y1=0 (A.U 2012)8. Find ( A.U 2013)9. Solve given Yo=3( A.U 2007)10. Derive the Difference equation from (A.U 2009)11. Using Z- transform solve given that Yo=3 and Y1=-5.(A.U 2010)

12. Prove that

13. State and prove final value theorem.(A.U 2008)14. State and prove Convolution theorem. on Z transform (A.U 2008)

15. Find using Convolution theorem.(A.U 2010)

16. Find Z and Z (A.U 2010 nov 2013)

17. Derive the difference equation from (A.U 2011)18. From , derive a difference equation by eliminating the constants.19. Derive the difference equation from .(A.U 2010)20. Solve the difference equation where y(0) = 1 and y(1) = 0.(A.U 2011)21. Using Z- Transform, solve .(A.U 2011)22. Using Z- Transform, solve given uo=1 and u1=2 (A.U 2009,2010)23.Using Z- Transform, solve given yo=0 =y1 (A.U nov 2013)24. Form the difference equation fromy(n)=(A+Bn) (Nov 2013)

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