Lesson Plan Sem 3

RENGANAYAGI VARATHARAJ COLLEGE OF ENGINEERINGDEPARTMENT OF MATHEMATICS

LESSON PLAN

Name of the Faculty: S.PAULSAMY Designation / Department: Professor/ MathematicsSubject Code : MA2211 Subject Name : Engineering Mathematics - III Year / Sem: II / III

Sl. No. DATELECTURE

TOPICOBJECTIVE

TIME REQUIRED

MODE MEDIACumulative

No. of Hours

BOOKS (REFERENCE /

TEXT)Unit – I Topic : Fourier series Target Period :

01 05.07.12Introduction

Knows the periodic function and Dirichlet conditions

50 Minutes LectureChalk & Board

1

Engineering Mathematics -III

Dr.A.Singaravelu, Meenakshi Agency.

02 06.07.12About Fourier series

To Knows the definition of Fourier series and problems under Fourier series.

50 MinutesLecture Chalk &

Board 2

03 09.07.12 Euler formulaTo prove Euler formula.

50 MinutesLecture

Chalk & Board 3

04 10.07.12 To find Fourier series expansion

Problems in the intervals (0,2l) and (0, 2 ).

50 Minutes Lecture Chalk & Board

4

1

05 11.07.12To find Fourier series expansion

Some problems to find expansion in the interval (-l,l) and (- , )

50 MinutesLecture

Chalk & Board 5

06 12.07.12To find Fourier series expansion

Some Problems under odd and even function in (- , ).


6

07 13.07.12Half range expansion

To solve problems under Fourier sine series in the interval (0,l) and (0, ).


7

08 16.07.12Half range expansion

To solve problems under Fourier cosine series in the interval (0,l) and (0, ).


8

09 17.07.12 Complex or exponential form of Fourier series

Problems under complex form


9

2

10 18.07.12RMS value and

Parseval’s identity

To knows the RMS value and To find Some Problems under parseval’s identity.


10

11 19.07.12Harmonic Analysis

Problems under Harmonic analysis


Board11

12 20.07.12Revision II Unit I

Tutorial 50 MinutesLecture Chalk &

Board12

Part – A1. Define periodic function with examples.2. When does a function posses a Fourier series expansion interms of trigonometric terms? (or) Explain Dirichlet’s conditions?3. Write the Fourier coefficients a0, an and bn in (0,2 ).4. Write the formula for Fourier constants of f(x) in (c, c + 2l).5. Can tan x be expanded in Fourier series. If so how? If not why?

6. Find the sum of the Fourier series of f(x) = x, 0 < x < 1

2, 1 < x < 2 , at x = 1.

7. Find the constant term in the Fourier series corresponding to f(x) = cos2x expressed in the interval (- , ). 8. Write a0, an in the expansion x + x3 as a Fourier series in (- , ). 9. Find the constant term in the Fourier series corresponding to f(x) = x - x3 in (- , ). 10. Find the constant term in the Fourier expansion of x2 – 2, in the range |x| 2.

3

11. If f(x) = x2 + x is expressed as a Fourier series in the interval (-2, 2) to which value this series converges at x = 2.

12. Find the Fourier constants bn for x sin x in (- , ). 13. Determine the value of an in the Fourier series expansion of f(x) = x3 in - < x <

14. If f(x) = 2x in the interval (0 , 4) then find the value of a2 in the Fourier series expansion.15. Expand f(x) = 1 in a sine series in 0 < x < .16. Expand f(x) = x in (0 , 1) as a Fourier sine series.17. Find the Fourier sine series of f(x) = x in 0 < x < 2.18. State Parseval’s identity for full range expansion of f(x) as Fourier series in (0 , 2l).19. If the Fourier series corresponding to f(x) = x in the interval (0 , 2 ) is

+ [an cosnx + bn sinnx] , with out finding the values of a0, an, bn find the value

of + [a + b ].

20. Define the RMS value of a function f(x) in (a , b).21. Define the RMS value of a function f(x) in (c , c + 2l).22. Define the RMS value of a function f(x) in (0 , 2 ).23. Find the RMS value of the function f(x) = x in the interval (0 , l).24. State Parseval’s identity for the half – range Cosine expansion of f(x) in (0 , l).25. What do you mean by Harmonic Analysis?

PART – B1. Find the Fourier series expansion of period 2l for the function f(x) = ( l –x)2

in the range (0, 2l). Deduce the sum of the series .

2. Find the Fourier series for f(x) = x, in 0 ≤ x ≤ 3

6-x, in 3 ≤ x ≤ 6. 3. Express f(x) = x sin x as a Fourier series in 0 ≤ x ≤ 2 . 4. Expand f(x) = x - x2 as a Fourier series in –L < x < L. 5. Obtain Fourier series for f(x) of period 2L and defined as follows.

f(x) = L + x, in (-L , 0)

4

L – x in (0 , L) . Hence deduce that + 23

1+

25

1+……. = .

6. Find the Fourier series for f(x) = -K, - < x < 0 K, 0 < x < . Hence deduce that

1 - + - +……. = .

7. Prove that for - < x < , = - + - ……

8. Obtain the Fourier series expansion of f(x) given by f(x) = 1+ , - ≤ x ≤

1- , 0 ≤ x ≤ and hence

deduce that + 23

1+

25

1+……. = .

9. Find the Fourier series for f(x) = |cos x| in the interval (- , ).

Obtain the sine series for the function f(x) = x, in 0 ≤ x ≤

-x, in ≤ x ≤

10. Find the half – range sine series of f(x) = 1 – x in (0 , 1).

11. Find the half – range sine series for f(x) = x ( - x) in (0 , ).

Deduce that - + -……. = .

12. Obtain a half – range cosine series of the function f(x) = kx, in 0 ≤ x <

k( -x), in ≤ x ≤ .

5

13. Find a cosine series for the function f(x) = x, in 0 ≤ x <

- x, in ≤ x < .

14. Obtain the Fourier expansion of x sin x as a cosine series in (0, ) and hence deduce the value of 1 + - + - ………...

15. Find the complex form of Fourier series f(x) = cos ax in - < x < .16. Find the complex form of Fourier series f(x) = in - < x < .

17. Find the Fourier series for f(x) = x2 in - < x < . Hence show that

+ + +……. = .

18. If for 0 < x < , the function f(x) has the expansion f(x) = , show that

19. The following table gives the variation of a periodic function over the period T.

x 0 T

f(x) 1.98 1.3 1.05 1.3 -0.88 -0.25 1.98

Show that f(x) = 0.75 + 0.37 cos + 1.004 sin , where = .

20. Find the Fourier series as far as the second harmonic to represent the function given in the following data.

x 0 1 2 3 4 5f(x) 9 18 24 28 26 20

ASSIGMENT TOPICS:1. Obtain the constant term and the first harmonic in the Fourier seriews expansion for f(x)

6

where f(x) is given in the following data.

x 0 1 2 3 4 5 6 7 8 9 10 11f(x) 18.0 18.7 17.6 15.0 11.6 8.3 6.0 5.3 6.4 9.0 12.4 15.7

TEXT BOOKS:

1.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition, Khanna Publications, Delhi, (2007).

REFERENCES:

1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 7th edition,Laxmi Publication (P) Ltd.,(2007).

2.Ramana B.V, “Higher Engineering Mathematics” ,Tata McGraw Hill Publishing Company,New Delhi, (2007).

3.Glyn james, “Advanced Engineering Mathematics” , 3rd edition, Pearson Education,(2007).

4. Erwin Kreyszig, “Advanced Engineering Mathematics” , 8th edition, Wiley india, (2007).


LESSON PLAN

Name of the Faculty: K.SUBHA Designation / Department: Lecturer / MathematicsSubject Code : MA2211 Subject Name : Engineering Mathematics - III Year / Sem: I / III

7

Sl. No. DATELECTURE

TOPICOBJECTIVE

TIME REQUIRED


No. of Hours

BOOKS (REFERENC

E / TEXT)Unit – I1 Topic : Fourier Transforms Target Period :

01 23.07.12Introduction

To know Definitions for Fourier integral transform and complex form of Fourier integrals.


1

Engineering Mathematics -III


02 24.07.12

Fourier sine and cosine integrals

To solve Problems based on Fourier sine and cosine integrals.

50 Minutes

Lecture Chalk & Board

2

03 25.07.12Fourier transform

To know Definition and properties

50 MinutesLecture

Chalk & Board 3

04 26.07.12

Fourier transform

To solve Problems based on Fourier transform

50 Minutes

LectureChalk & Board

4

05 27.07.12 Convolution of two functions in Fourier transform

To solve Problems based on convolution theorem.


5

8

06 30.07.12Parseval’s identity

To solve problems based on Parseval’s identity.


6

07 31.07.12Fourier sine and cosine transforms

To know definition for Fourier sine and cosine transform.


7

08 01.08.12

Properties of Fourier sine and cosine transforms

To know the properties of Fourier sine and cosine transform.


8

09 02.08.12Inversion formula

To solve problems based on Fourier cosine transform.


9

10 03.08.12Fourier sine transform

To solve problems based on Fourier sine transform


10

11 06.08.12Revision I Unit I


Board11



Board12

PART – A1. State Fourier integral theorem.2. Show that f(x) = 1, 0 < x < can not be represented by a Fourier integral.3. Define Fourier transform pair. (or) Define Fourier transform and its inverse transform.4. If Fourier transform of f(x) = F(s) then what is Fourier transform of f(ax).

9

5. Find the Fourier transform of f(x) = 1 for |x| < a

0 for |x| > a > 0 6. What is the Fourier cosine transform of a function. Write down the Fourier cosine transform pair of formulae.

7. Find the Fourier cosine transform of f(x) = cos x if 0 < x < a 0 if x a

8. Find the Fourier cosine transform of e-x.9. Find the Fourier cosine transform of e-3x.10. Find the Fourier sine transform of e-x.11. Find the Fourier sine transform of e-3x.12. Find the Fourier sine transform of 3e-2x.

13. Find the Fourier sine transform of .

14. Find Fc[xe-ax] and Fs[xe-ax].

15. Prove that Fc[f(x)cosax] = [f(s+a) + f(s-a)] where Fc denote the Fourier cosine

transform of f(x).17. IF F(s) is the Fourier transform of f(x), then show that the Fourier transform of eiaxf(x) is F(s+a).

18. Given that e- is self reciprocal under Fourier cosine transform, find Fourier Sine transform of x e- .

19. If Fc(s) is the Fourier cosine transform of f(x). prove that the Fourier cosine transform of f(ax) is Fc( ).

20. If F(s) is the Fourier transform of f(x), then find the Fourier transform of f(x-a).21. If Fs(s) is the Fourier sine transform of f(x),

Show that Fs[f(x)cosax] = [Fs(s+a) + Fs(s-a)].

22. Find F[xnf(x)] in terms of Fourier transform of f(x).

23. Find F[ ]in terms of Fourier transform of f(x).

10

24. State the Convolution theorem for Fourier transforms.25. State Parseval’s identity for Fourier transform.

PART – B

1. Show that the Fourier transform of f(x)= a2-x2, |x|< a 0, |x| > a is

. Hence deduce that .

Using Parseval’s identity show that = .

2. Find the Fourier transform of f(x) = 1 – x2, |x| < 1 0 , |x| > 1.

Hence prove that .

3. Find the Fourier transform of f(x) = 1 , |x| < a 0, |x| > a > 0.

Hence deduce that (i) (ii) .

4. Find the Fourier transform of f(x) = 1-|x|, |x| < 1 0 , |x| > 1.

Hence deduce that and .

5. Find the Fourier cosine and sine transform of the function f(x) = e-ax , a > 0.

6. Evaluate , if a > 0 using parseval’s identity.

7. Evaluate , if a > 0 using parseval’s identity.

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8. Evaluate , if a, b > 0 using parseval’s identity.

9. Evaluate , if a, b > 0 using parseval’s identity.

10. Evaluate , using Transform method.

11. Find the Fourier cosine and sine transform of f(x) = xe-ax.

12. Find the Fourier cosine transform of f(x) = .

13. Find the Fourier sine transform of f(x) = .

14. Find the Fourier cosine transform of f(x) = .

15. Find the Fourier cosine transform of , a > 0 and hence deduce that sine transform of x .

16. Show that the Fourier sine transform of x is self reciprocal.

17. Find the Fourier sine and cosine transform of e-x also find the Fourier sine transform of and Fourier cosine transform of .

18. State and prove Convolution theorem in Fourier transform.

ASSIGMENT TOPICS:

1.Find the Fourier transform of f(x) = a- |x|, |x| < a 0, |x| > a > 0.

Deduce that (i) .

TEXT BOOKS:

1.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition, Khanna Publications, Delhi, (2007).

REFERENCES:

12

1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 7th edition,Laxmi Publication (P) Ltd.,(2007).





LESSON PLAN

Name of the Faculty: Dr.S.Paulsamy, K.Subha, N.Ananthan Designation / Department: Professor/Lecturer/ Lecturer / Mathematics

Subject Code : MA2111 Subject Name: Engineering Mathematics -II Year / Sem: I / II

13

Sl. No. DATELECTURE

TOPICOBJECTIVE

TIME REQUIRED


No. of Hours

BOOKS (REFERENCE /

TEXT)Unit – I11 Topic :ANALYTIC FUNCTIONS Target Period :

01 07.02.12Complex Variables

To know the properties of Arithmetic Operations and limit to the functions


1

Engineering Mathematics -II


02 08.02.12Functions of a Complex Variables

To know the Continuity of a Function and some problems for that Continuity functions


Board 2

03 14.02.12

The necessary condition for f(z) to be analytic

To know the Cauchy Riemann equations and to find some problems

50 MinutesLecture

Chalk & Board 3

04 15.02.12 Cauchy – Riemann Equations

Using Cauchy Riemann equations to find some problems


4

14

05 21.02.12Sufficient condition of f(z) to be analytic

To know the sufficient conditions and Polar form of Cauchy Riemann equations and to find some problems

50 MinutesLecture

Chalk & Board 5

06 22.02.12 Properties of analytic functions

To know some properties and using to find some problems


6

07 28.02.12Constructions of analytic functions

To know the Milne – Thomson method and using some problems


7

08 29.02.12 Harmonic Functions

To know the Harmonic functions and using some problems


8

15

09 13.03.12Conformal Mappings

To know what is Conformal mapping and using Transformations to find some problems


9

10 14.03.12Bilinear Transformation

To known what is bilinear transformations and to find some problems


10



Board11



Board12

PART –A1.State the Cauchy – Riemann equation in polar coordinates.2.Give an example that the Cauchy – Riemann equation are necessary but not sufficient for a function to be analytic..3.State the sufficient condition for the function f(z) to be analytic.4.If w = ez, find dw/dz using complex variable.5. Choose the correct answer, w = f(z) is analytic function of z,then------------.6. The function f(z) = u+iv is analytic only if ----------7. Show that the function f(z) = z z¯ is not analytic at z = 0.8. State the necessary and sufficient condition for f(z) to be analytic.9. Verify whether w = sin xcos hy+ i cos x sin hy is analytic or not.10. The function f(z) = √׀xy׀ is not regular at the origin.11. The function w = log z is analytic every where in the complex plane.12. A certain function u(x,y) can be the real part of an analytic function if----------.13. If Cauchy Riemann equation are satisfied by real and imagionary parts of a complex function f(z) then f(z) should be analytic.

16

14. Write down the necessary condition for w = f(z) = f(reiɵ) to be analytic.15. State two properties of analytic function.16. Show that the function x4-6x2y2+y4 is harmonic.17. Examine whether the function xy2 can be real part of an analytic function.18. Define Mobius transformation.19. Define isogonal transformation.20. What is invariant point in a mapping?PART –B 1 .a. Find the analytic function f(z) whose real part is u(x,y) = 3x2y+2x2 – y3 – 2y2

b. Find the bilinear transformation that maps 1,i,-1 of the z- plane onto 0,1,∞ of the w – plane.2 .a. Find the image of 1 < x < 2 under the mapping w = 1/z. b. If u+v = (x-y)(x2+4xy+y2) and f(z) = u+iv, find f(z) interms of z.3.a. Show that the transformation w = z-i/ 1-iz maps (i) the interior of the circle׀ z 1= ׀ on to the lower half of the w- plane and (ii) the upper half of the z-plane on to the interior of the circle ׀w 1= ׀ . b. Find the analytic function u+iv, if u = (x-y)(x2+4xy+y2). Also find the conjugate harmonic function v.4. a. Find the image of the hyperpola x2-y2 = 1 under the transformation w = 1 / z. b. Prove that the transformation w = z / 1-z maps the upper half of z – plane onto the upper half of w-plane. What is the image of ׀z 1 =׀ under this transformation?.ASSIGMENT TOPICS:1. If a,b, are the two fixed points of a bilinear transformation, show that it can be written in the form, w-a/w-b = K(z-a/z-b), K is Constant, a ≠ b.2. If f(z) is a regular function of z, prove that (∂2/∂x2 + ∂2/∂y2) ׀f(z) ׀4 = 2׀ f’(z) 2׀ .

TEXT BOOKS:

1.Bali N.P and Manish Goyal, , “Text Book of Engineering Mathematics” , 3rd edition,Laxmi Publication (P) Ltd.,(2008).

2.Grewal. B.S, “Higher Engineering Mathematics” , 40th edition,Khanna Publications, Delhi, (2007).

REFERENCES:



17


2.Jain R.K and Iyengar S.R.K, “Advanced Engineering Mathematics” , 3rd edition,Narosa Publishing House Pvt.Ltd.,(2007).


LESSON PLAN

Name of the Faculty: Dr.S.Paulsamy, K.Subha, N.Ananthan Designation / Department:Professor/Lecturer/ Lecturer / MathematicsSubject Code : MA2111 Subject Name: Engineering Mathematics- II Year / Sem: I / II

Sl. No. DATELECTURE

TOPICOBJECTIVE

TIME REQUIRED


No. of Hours

BOOKS (REFERENC

E / TEXT)Unit – 1V Topic : COMPLEX INTEGRATION01 03.04.12 Properties of

complex integrals

To know the some properties and to find some problems


1 Engineering Mathematics -II

18


02 04.04.12Cauchy’s integral theorem

To know the statement of Cauchy’s Integral theorem and to find some problems


Board 2

03 10.04.12Cauchy’s integral formula

Some problems using to find Cauchy’s Integral formula for derivatives

50 MinutesLecture

Chalk & Board 3

04 11.04.12 Taylor Series

To know some important results and using to find some problems

50 MinutesLecture

Chalk & Board 4

05 17.04.12 Laurent’s Series

To know the Laurent’s Series expantions and to find some problems

50 MinutesLecture

Chalk & Board 5

06 18.04.12 Singularities To know types of singularities and using to find some problems


6

19

07 02.05.12Residue at a Pole

To know definitions for Residue and Pole and using to find some problems


7

08 04.05.12Cauchy’s Residue theorem

To know the Statement of Cauchy’s Residue theorem and using to find some problems


8

09 09.05.12

Evaluation of integrals using Rasidue theorem

Some problems using to find Residue Theorem


9

10 14.05.12

Real definite integrals by contour integration

To know the Definition for Definite integral and Contour integrations and using to find some problems


10



Board11



Board12

20

PART - A1.What is the value of ∫ 3z2+7z+1/ z+1 if C is ׀z׀= ½ over the region C. 2. State Cauchy’s Integral Formula.3. Find the value of ∫ z / z2-1 where C is ׀ z׀ = ½ over the region C.4. State Cauchy’s Integral Formula.5. The integral value of ∫ (x2 – iy) dz along the path y = x is 1/6(5-i) and the limits from 0 to 1+i say true or false?.6. Find the Laurent’s expansion of f(z) = 1/ z(1-z)2 in ׀z 1 >׀ .7. Expand cos z in a Taylor’s series at z = π / 4.8. Expand the function sin z / z-π about z = π9. Laurent’s series expansion of f(z) over a given annular is not unique, say true or false?.10. Expand 1 / z-2 at z = 1 as a Taylor’s series.11. What are the poles of cot z?12. Define essential singular point of f(z)?.13. Find the zero’s of z3-1 / z3+114. If f(z) = 1 / z2+1 is analytic everywhere except ----------15. For a simple pole at z = a the residue of f(z) at z = a, where f(z) = P(z) / Q(z) is ---------16. Evaluate ∫ cos π z / z-1 dz where C is ׀z 1.5 = ׀ over the region C.17. The residue at the pole of the function z / z2 +1 are equal.18. If f(a) = ∫ z2+1 / z-a dz where C is the ellipse x2 + 4y2 = 4, then the value of f(1) is ----------------, over the region C.19. State the function f(z) and the region contour to evaluate the integral ∫ x sin x / x4 + a4 dx, and the limits from 0 to ∞.20. Express ∫ dɵ / 1+ a cos ɵ as a contour integral around the circle ׀z 1 = ׀ , and the limits from 0 to 2π.

PART –B1.Evaluate ∫ dx / (x2+a2)3, a > 0, using contour integrtion, and the limits from 0 to ∞.2.Evaluate ∫ dɵ / 2+ cos ɵ , and the limits from 0 to 2π.3. Expand 1 / (z-1)(z-2) in Laurent’s series valid for ׀ z 1 <׀ and 1<׀ z 2 < ׀ .4.Evaluate ∫ dɵ / a+b cos ɵ where a>b by contour integration, and the limits from 0 to 2π.5.Using residue calculus prove that ∫ dɵ / 5+3 cos ɵ = π / 2 and the limits from 0 to ∞.

ASSIGMENT TOPICS:

1.Evaluate ∫x2dx / (x2+a2)(x2+b2) using contour integration, where a > b> ɵ, and the limits from -∞to ∞.

2.Evaluate ∫ dɵ / 2-cos ɵand the limits from 0 to 2π.

21

TEXT BOOKS:



REFERENCES:






LESSON PLAN

Name of the Faculty:Dr. S.Paulsamy, K.Subha, N.Ananthan Designation / Department:Professor/Lecturer/ Lecturer / MathematicsSubject Code : MA2111 Subject Name : Engineering Mathematics -II Year / Sem: I / II

Sl. No. DATELECTURE

TOPICOBJECTIVE

TIME REQUIRED


No. of Hours

BOOKS (REFERENC

E / TEXT)

22

Unit – V Topic : LAPLACE TRANSFORM

01 02.04.12

Conditions for existence of Laplace transforms

Know that Laplace definition and some problems based on exponential orderand sine, cosine method


1

Engineering Mathematics -II


02 04.04.12 Linear Property

Know that the linear property definition and solve some problems


Board 2

03 09.04.12First Shifting Theorem

Know that First shifting theorem and solve some problems

50 MinutesLecture

Chalk & Board 3

04 11.04.12 Second Shifting Theorem

Know that Second shifting theorem and solve some problems


4

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05 12.04.12

Laplace transforms of Derivatives And integrals

Know that the derivative and integral theorem, Dirac delta function and solve some problems

50 MinutesLecture

Chalk & Board 5

06 16.04.12

Laplace Transform of periodic functions

Know that the periodic functions and solve some problems


6

07 18.04.12The Inverse laplace transform

Know that some formula for inverse Laplace transform and solve some problems


7

08 25.04.12Convolution Theorem

Know that Convolution theorem and using some problems


8

09 26.04.12 Initial value Problem and Final Value Theorem

Solve some problems based on initial value problem and final value theorem


9

24

10 27.04.12

Laplace Transforms for differential equations

Solve some problems based on differential equations


10

11 30.04.12Simultaneous Differential Equations

Solve some problems based on simultaneous differential equations


11



Board12



Board13

PART –A

1. If L(sin at / t) = cot -1 (s / a) then ∫ sin at / t dt = ---------, and the limits from 0 to ∞.2. Say true or false L[t(f(t)] = Lf(t) / s

3. L[f(t)] = ϕ(s) then L[f (t/2)] = 2ϕ(2s).

4. Find the Laplace transform of cos3t / t.

5. Find the Laplace transform ׀ sin wt׀.

6. Find L[sin √t].

7. If L [f (t)] = s2-s+1 / (2s+1)2(s-1), applying the change of scale property shown that L [f (2t)] = s2-2s+1 / 4(s+1)2(s-2).

8. L (u (t-a)) = ------------ (or) Give the Laplace transform of unit step function.

9. Find the Laplace transform of Dirac Delta Function.

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10. If f(t) = e-2t sin2t, find L[f’(t)]

11. L[f(t)] = 1/ s(s+1) = f(s) find f(0) and f(∞).

12. Find L-1(1 / s2+4s+4).

13. Find L[cos √t / √t]

14. Find the inverse transform of tan-1s.

15. Find L-1[1 / s4-1]

16. Find L-1(1 / s2(s2+81)).

17. Find L-1(e-πs / s2)

18. Solve the integral equation.

19. Solve y+ ∫ydt = t2+2t and the limits from 0 to t.

20. Show that [1*1*1*……*1] n times = tn-1 / (n-1)!.

PART – B 1. Solve the initial value problem. y” – 3y’+2y = 4t, y(0) = 1, y’(0) = -1.2. Using convolution theorem, find L-1[s / (s2+1)2(s2+4)].3.a. Find L(e-2t t sin2t), b. find L-1[tan-1(2 / s2)]4. Using convolution theorem find the inverse Laplace transform of 1 / (s2+4)2.5. Find L-1(s/ s4+4a4).

ASSIGMENT TOPICS:1 Find the Laplace transform of the following triangular wave function given by f(t) = { t, 0 ≤ t ≤ π and 2-t, π ≤ t ≤ 2π f(t+2) = f(t).

2.Verify initial and final value theorems for the function f(t) = 1+e-t(sin t + cos t).

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TEXT BOOKS:



REFERENCES:





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Lesson Plan Sem 3

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Transcript of Lesson Plan Sem 3