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Show that the estimates of the population variance from the samples are ... same group of students to find the number of
Engineering Mathematics

2015

SUBJECT NAME

: Statistics and Numerical Methods

SUBJECT CODE

: MA6452

MATERIAL NAME

: Additional Problems

MATERIAL CODE

: JM08AM1015

REGULATION

: R2013

UPDATED ON

: March 2015

Name of the Student:

Branch:

Unit – I (Testing of Hypothesis)  Students ‘t’ test 1) The following data gives the lengths of 12 samples of Egyptian cotton taken from a large consignment. 48, 46, 49, 46, 52, 45, 43, 47, 47, 46, 47, 50. Test if the mean length of the consignment be taken as 46. 2) A random sample of size 16 valves from a normal population showed a mean of 53 and a sum of squares of deviation from the mean equal to 150. Can this sample by regarded as taken from the population having 56 as mean? Obtain 95% confidence limits of the mean of the population. 3) Two random samples gave the following results. Sample Size Sample Mean Sum of Squares of deviations from the mean 1

10

15

90

2

12

14

108

Test whether the samples come from the same normal population. 4) Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results. Horse A

28

30

32

33

33

29

Horse B

29

30

30

24

27

29

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

34

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2015

Test whether the two horses have the same running capacity. 5) To verify whether a course in accounting improved performance, a similar test was given to 12 participant both before and after the course. The marks are Before

44

40

61

52

32

44

70

41

67

72

53

72

After

53

38

69

57

46

39

73

48

73

74

60

78

Was the course useful?

 F – test 1) For a random sample of 10 pigs fed on diet A the increases in weight in a certain period were : 10, 6, 16, 17, 13, 12, 8, 14, 15, 9. For another random sample of 12 pigs, fed on diet B, the increases in the sample period were: 7, 13, 22, 15, 12, 14, 18, 8, 21, 23, 10, 17. Show that the estimates of the population variance from the samples are not significantly different. 2) The random samples were drawn from two normal populations and the following results were obtained. Sample I

16 17 18 19 20 21 22 24 26 27

Sample II 19 22 23 25 26 28 29 30 31 32 35 36 Obtain estimates of the variances of populations and test whether the two populations have the same variances. 3) In one sample of 10 observations from a normal population, the sum of the squares of the deviations of the sample values from the sample mean is 102.4 and in another sample of 12 observations from another normal population, the sum of the squares of the deviations of the sample values from the sample mean is 120.5. Examine whether the two normal populations have the same variances. 4) Pumpkins were grown under two experimental conditions. Two random samples of 11 and 9 pumpkins show the sample standard deviations of their weights as 0.8 and

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

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0.5 respectively. Assuming that the weight distributions are normal, test hypothesis that the true variances are equal.

 Chi-Square test (Goodness of fit) 1) The number of automobile accidents per week in a certain community are as follows: 12, 8, 20, 2, 14, 10, 15, 6, 9, 4. Are these frequencies in agreement with the belief that accident conditions were the same during this 10 week period. 2) The following figures show the distribution of digits in numbers chosen at random from a telephone directory. Digits

0

1

2

Frequency 1026 1107 997

3

4

5

6

7

966 1075 933 1107 972

8

9

964

853

Test whether the digits may be taken to occur equally frequently in the directory. 3) A die is thrown 264 times with the following results. Show that the die is biased. [Given] 𝜓0.05 = 11.07 𝑓𝑜𝑟 5 𝑑. 𝑓] No. appeared on 1 2 the die Frequency

40

32

3

4

5

6

28

58

54

52

4) A sample analysis of examination results of 500 students was made. It was found that 220 students had failed, 170 had secured a third class, 90 were placed in second class and 20 got a first class. Do these figures commensurate with the general examination result which is in the ratio of 4:3:2:1 for the various categories respectively.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

 Chi-Square test (Independence of attributes) 5) In an experimental on immunization of cattle from tuberculosis the following research were obtained. Not affected Affected Inoculated

12

26

Not Inoculated

16

6

Calculate 𝜓 2 and discuss the effect of vaccine in controlling susceptibility to tuberculosis. 6) Given the following contingency table for hair colour and eye colour. Find the value of 𝜓 2 . Is there good association between the two.

Fair

Hair Colour Brown

Black

Blue

15

5

20

40

Grey

20

10

20

50

Brown

25

15

20

60

Total

60

30

60

150

Total

Eye Colour

7) Two researchers adopted different sampling techniques while investigating the same group of students to find the number of students falling into different intelligence level. The results are as follows.

Researchers

Below Average

Average

Above Average

Genius

Total

X

86

60

44

10

200

Y

40

33

25

2

100

Total

126

93

69

12

300

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Would you say that the sampling techniques adopted by the two researchers are significantly different [Given 5% value of 𝜓 2 for 2 degrees and 3 degree freedoms are 5.991 and 7.82 respectively] 8) 1000 students at college level were graded according to their I.Q. and the economic conditions of their homes. Use 𝜓 2 - test to find out whether there is any association between economic conditions at home and I.Q. I.Q. Economic conditions

High

Low

Total

Rich

460

140

600

Poor

240

160

400

Total

700

300

1000

9) In a certain sample of 2000 families 1400 families are consumers of tea. Out of 1800 Hindu families, 1236 families consume tea. Use  2 test and state whether there is any significant difference between consumption of tea among Hindu and Non-Hindu families.

 Large Sample (n > 30) Single Mean & Difference of Means: 1) The mean lifetime of a sample of 100 light tubes produced by a company is found to be 1580 hours with standard deviation of 90 hours. Test the hypotheses that the mean lifetime of the tubes produced by the company is 1600 hours. 2) The Mean breaking strength of the cables supplied by a manufacturer is 1800 with an SD of 100. By a new technique in the manufacturing process, it is claimed that the breaking strength of the cable has increased. To test this claim a sample of 50 cables is tested and is found that the mean breaking strength is 1850. Can we support the claim at 1% level of significance.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

3) A simple sample of heights of 6400 English men has a mean of 170 cm. and a S.D of 6.4 cm, while a simple sample of heights of 1600 Americans has a mean of 172 cm. and a S.D of 6.3 cm. Do the data indicate that Americans are the average taller than the English men?

Single Proportion & Difference of Proportions: 4) In a sample of 400 parts manufactured by a factory, the number of defective parts was found to be 30. The company, however, claimed that only 5% of their product is defective. Is the claim tenable? 5) 40 People were attacked by a disease and only 36 survived. Will you reject the hypothesis that the survival rate, if attacked by this disease, is 85% in favour of the hypothesis that it is more at 5% level of significance. 6) In a sample of 1000 people in Karnataka 540 are rice eaters and the rest are wheat eaters. Can we assume that both rice and wheat are equally popular in this sate at 1% level of significance? 7) Random samples of 400 men and 600 women were asked whether they would like to have a flyover near their residence. 200 men and 325 women were in favour of the proposal. Test the hypothesis that proportions of men and women in favour of the proposal are same, at 5% level. 8) In a sample of 600 students of a certain college 400 are found to use dot pens. In another college, from a sample of 900 students 450 were found to use dot pens. Test whether the two colleges are significantly different with respect to the habit of using dot pens.

Unit – II (Design of Experiments)  Completely Randomized Design (One – Way Classification) 1) A random sample is selected from each of three makes of ropes and their breaking strength (in pounds) are measured with the following results. I 70 72 75 80 83 II 100 110 108 112 113 120 107 III 60 65 57 84 87 73 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

Test whether the breaking strength of the ropes differs significantly. 2) The following are the number of mistakes made in 5 successive days by 4 technicians working for a photographic laboratory test at a level of significance α = 0.01. Test whether the difference among the four sample means can be attributed to chance.

Technician I

II

III

IV

6

14

10

9

14

9

12

12

10

12

7

8

8

10

15

10

11

14

11

11

3) As part of the investigation of the collapse of the roof of a building, a testing laboratory is given all the available bolts that connected a the steel structure at three different positions on the roof. The forces required to shear each of these bolts (coded values) are as follows: Position 1 90

82 79 98

83 91

Position 2 105 89 93 104 89 95 86 Position 3 83

89 80 94

Perform an analysis of variance to test at the 0.05 level of significance whether the differences among the sample means at the three positions are significant.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

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4) A completely randomized design experiment with ten pots and three treatments gave the results given below. Analyze the results for the effects of treatments.

Treatment Replications A

5 7 1 3

B

4 4 7

C

3 1 5

 Randomized Block Design (Two – Way Classification) 5) The following data represent the number of units of production per day turned out by different workers using 4 different types of machines.

Machine type A

B

D

C

1 44 38 47 36 Workers 2 46 40 52 43 3 34 36 44 32 4 43 38 46 33 5 38 42 49 39 6) A company appoints 4 salesmen A,B,C and D and observes their sales in 3 seasons: summer, winter and monsoon. The figures (in lakhs of Rs.) are given in the following table:

Salesmen Season

A

B

C

D

Summer 45 40 38 37 Winter

43 41 45 38

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

Monsoon 39 39 41 41 Carry out an analysis of variance. 7) Four different, though supposedly equivalent forms of a standardized reading achievement test were given to each of 5 students, and the following are the scores, which they obtained. Student 1 Student 2 Student 3 Student 4 Student 5 Form A

75

73

59

69

84

Form B

83

72

56

70

92

Form C

86

61

53

72

88

Form D

73

67

62

79

95

Perform a two-way analysis of variance to test at the level of significance α = 0.01 8) An experiment was performed to judge the effect of 4 different fuels and 3 different types of launchers on the range of a certain rocket. Test on the basis of the following ranges, in miles, whether there is a significant effect due to differences in fuels and whether there is a significant effect due to difference in launchers. Fuel 1 Fuel 2 Fuel 3 Fuel 4 Launcher X

45.9

57.6

52.2

41.7

Launcher Y

46.0

51.0

50.1

38.8

Launcher Z

45.7

56.9

55.3

48.1

Use 0.01 level of significance.

9) An experiment was designed to study the performance of 4 different detergents for cleaning fuel injectors. The following ‘cleanness’ readings were obtained with specially designed equipment for 12 tanks of gas distributed over 3 different models of engines:

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

Detergent Engine 1 Engine 2 Engine 3 Total A

45

43

51

139

B

47

46

52

145

C

48

50

55

153

D

42

37

49

128

Total

182

176

207

565

Test at the 0.01 level of significance whether there are differences in the detergents or in the engines. 10) A laboratory technician measures the breaking strength of each of 5 kinds of linen threads by using 4 different measuring instruments, and obtains the following results in ounces.

I1

I2

I3

I4

Thread 1 20.9 20.4 19.9 21.9 Thread 2

25

26.2 27.0 24.8

Thread 3 25.5 23.1 21.5 24.4 Thread 4 24.8 21.2 23.5 25.7 Thread 5 19.6 21.2 22.1 22.1 Perform a 2-way ANOVA using the 0.05 level of significance for both tests.

 Latin Square (Three – Way Classification) 11) Set up the analysis of variance for the following results of a Latin Square Design. (use α = 0.01) level of significance.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

A

C

B

D

12

19

10

8

C

B

D

A

18

12

6

7

B

D

A

C

22

10

5

21

D

A

C

B

12

7

27

17

12) Analyze the variance in the Latin square of yields (in kgs) of paddy where P,Q,R,S denote the different methods of cultivation. S122

P121

R123 Q122

Q124 R123

P122

S125

P120 Q119

S120

R121

R122

Q121 P122

S123

Examine whether the different methods of cultivation have given significantly different yields.

13) The figures in the following 5 X 5 Latin square are the numbers of minutes, engines E1 , E2 , E3 , E4 and E5 , tuned up by mechanics

M1, M 2 , M 3 , M 4 and M5 , ran with a gallon of fuel A,B,C,D and E.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

E1

E2

E3

E4

E5

M1

A31

B24

C20

D20

E18

M2

B21

C27

D23

E25

A31

M3

C21

D27

E25

A29

B21

M4

D21

E25

A33

B25

C22

M5

E21

A37

B24

C24

D20

Use the level of significance α = 0.01 to test the following a) The null hypothesis H 0 that there is no difference in the performance of the five engines. b) H 0 that the persons who tuned up these engines have no effect on their performance. c) H 0 that the engines perform equally well with each of the fuels. 14) In a 5 X 5 Latin square experiment, the data collected is given in the matrix below. Yield per plot is given in quintals for the fie different cultivation treatments A,B,C,D and E. Perform the analysis of variance. A48 E66

D56 C52 B61

D64 B62 A50 E64

C63

B69 A53 C60 D61 E67 C57 D58 E67 E67

B65 A55

C57 B66 A60 D57

15) In a Latin square experiment given below are the yields in quintas per acre on the paddy crop carried out for testing the effect of five fertilizers A,B,C,D,E. Analyze the data for variations.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

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B25 A18 E27

D30 C27

A19 D31 C29 E26

B23

C28 B22 D33 A18 E27 E28

C26 A20 B25 D33

D32 E25

B23 C28 A20

Unit – III (Solution of Equations and Eigenvalue Problems)  Newton – Raphson method 1) Solve for a positive root of the equation x 4  x  10  0 using Newton – Raphson method. 2) Compute the real root of x log10 x  1.2 correct to three decimal places using Newton-Raphson method. 3) Find the approximate root of xe x  3 by Newton – Raphson method correct to three decimal places. 4) Find an iterative formula to find the reciprocal of a given number N and hence find 1 the value of . 19 5) Find an iterative formula for N by Newton-Raphson method and hence find the approximate value of 142 .

 Gauss – Jacobi method & Gauss – Seidel method 1) Solve the following equation using Jacobi’s iteration method 20 x  y  2z  17; 3 x  20 y  z  18; 2 x  3 y  20z  25 . 2) Solve, by Gauss-Seidel method, the following system: 28 x  4 y  z  32; x  3 y  10z  24; 2 x  17 y  4z  35 . 3) Use Gauss – Seidel iterative method to obtain the solution of the equations: 9 x  y  2z  9; x  10 y  2z  15; 2 x  2 y  13z  17 .

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

 Gauss Elimination method & Gauss Jordan method 1) Solve the system of equations by Gauss – elimination method 10 x  2 y  3z  23; 2 x  10 y  5z  33; 3 x  4 y  10z  41 . 2) Solve by Gauss – elimination method 3 x  y  z  3; 2 x  8 y  z  5; x  2 y  9z  8 . 3) Solve x  3 y  3z  16; x  4 y  3z  18; x  3 y  4z  19 by Gauss – Jordan method.

 Matrix inversion by Gauss Jordan method  0 1 2 1) Find the inverse of the matrix by Gauss – Jordan method:  1 2 3  .  3 1 1    2 2 3 2) Using Gauss – Jordan method find the inverse of the matrix  2 1 1  .  1 3 5  

 Eigenvalues of a matrix by Power method 1) By using power method find the numerically largest eigenvalue and eigenvector of  15 4 3  the matrix  10 12 6  .  20 4 2     1 1 2  2) Find the dominant eigenvalues and eigenvector of A   0 3 1  , using power  0 0 5   method.  2 0 1 3) Find the largest eigenvalues and eigenvector of the matrix A   0 2 0  , using  1 0 2   power method.

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

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Unit – IV (Interpolation, Numerical Differential & Integration)  Lagrange’s & Newton’s divided difference interpolation 1) Using Lagrange’s formula, fit a polynomial to the data. 1 3 4 x 0 y -12 0 6 12 also find y at x  2 . 2) Use Lagrange’s formula to find a polynomial which takes the values f (0)  12, f (1)  0, f (3)  6 and f (4)  12 . Hence find f (2) .

3) Using Lagrange’s interpolation formula, find the equation of the cubic curve that passes through the points  1, 8  ,  0, 3  ,  2,1 and  3, 2  . 4) Find the value of x when y  85 , using Lagrange’s formula from the following table.

x y

2 5 8 14 94.8 87.9 81.3 68.7

5) Given the following data, find y(6) and the maximum value of y (if it exists)

x: 0 2 3 4 7 9 y : 4 26 58 112 466 922 6) Find f (1), f (5) and f (9) using Newton’s divided difference formula from the following data. x y  f ( x)

0 2 3 4 7 8 4 26 58 112 466 668

7) Find the cubic polynomial from the following table using Newton’s divided difference formula and hence find f (4) .

x y  f ( x)

0 1 2 5 2 3 12 147

8) Determine f ( x ) as a polynomial in x for the following data, using Newton’s divided difference formulae. Also find f (2) .

x

y  f ( x)

–4 –1 0 2 5 1245 33 5 9 1335

9) Find the function f ( x ) from the following table using Newton’s divided difference formula: Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

x:

0 1 2 4 5 7 f ( x ) : 0 0 –12 0 600 7308

 Newton’s forward and backward difference interpolation 1) The population of a town is as follows:

x Year: 1941 1951 1961 1971 1981 1991 y Population in 20 24 29 36 46 51 thousands: Estimate the population increase during the period 1946 to 1976. 2) Given the following table, find the number of students whose weight is between 60 and 70 lbs: Weight (in lbs) x : 0 – 40 40 – 60 60 – 80 80 – 100 100 – 120 No. of students: 250 120 100 70 50

 Approximation of derivatives using interpolation 1) Find the first two derivatives of x1/ 3 at x  50 and x  56 , for the given table:

x y  x1/ 3

50 51 52 53 54 55 56 3.6840 3.7084 3.7325 3.7563 3.7798 3.8030 3.8259

2) Find the first, second and third derivatives of the function tabulated below at the point x  1.5 .

x:

1.5

2.0 2.5

3.0

3.5

4.0

f ( x ) : 3.375 7.0 13.625 24.0 38.875 59.0

3) Find the first and second derivative of the function tabulated below at x  0.6 .

x

0.4

0.5

0.6

0.7

0.8

y

1.5836 1.7974 2.0442 2.3275 2.6511

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

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 Numerical integration using Trapezoidal & Simpson’s Rule 1

dx , using Trapezoidal rule with h  0.2 . Hence determine the value 1  x2 0

1) Evaluate  of  .

6

1 dx by using (i) direct integration (ii) Trapezoidal rule (iii) 1 x 0

2) Evaluate I  

Simpson’s one-third rule (iv) Simpson’s three-eighth rule. 1

3) Evaluate  e  x dx by dividing the range of integration into 4 equal part using 2

0

Trapezoidal rule and Simpson’s rule. 4) The velocity  of a particle at a distance S from a point on its path is given by the table below: S (meter) 0 10 20 30 40 50 60  (m / sec) 47 58 64 65 61 52 38 Estimate the time taken to travel 60 meters by Simpson’s 1/3rd rule and Simpson’s 3/8th rule.

 Double integrals by Trapezoidal and Simpsons’s rules 2 2

1.

Using Trapezoidal rule, evaluate

dxdy numerically with h  0.2 along x 2  y2

 x 1 1

direction and k  0.25 along y -direction. 1 1

2.

Evaluate

(M/J 2012)

1

  1  x  y dxdy by trapezoidal rule.

(N/D 2014)

0 0

1.2 1.4

3.

Evaluate

 1 1

dxdy by trapezoidal formula by taking h  k  0.1 . x y

2 1

4.

Evaluate

1

(A/M 2010)

1

  4 xy dxdy using Simpson’s rule by taking h  4 and k  2 .(N/D 2012) 0 0

1/ 2 1/ 2

5.

Evaluate

 0

0

1 sin( xy ) dxdy using Simpson’s rule with h  k  . 4 1  xy (M/J 2012),(M/J 2014)

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Engineering Mathematics 1.2 2.4

6.

Evaluate

 1 2

2015

1 dxdy using Simpon’s one-third rule. xy

(M/J 2013)

Unit – V (Numerical Solution of Ordinary Differential Eqns.)  Taylor’s series, Euler’s and Modified Euler’s method dy  x 2 y  1 and y(0)  1 . Find the value of y(0.1) using Taylor series dx method. dy 2) Find the Taylor series solution of y(0.1) given that  y 2  e x ; y(0)  1 . Compute dx using first five terms. dy y 3) Given  3 x  and y(0)  1 . Find the value of y(0.1) and y(0.2) using Taylor dx 2 series method. 4) Using Taylor series method, compute the value of y(0.2) correct to 3 decimal places

1) Given

dy  1  2 xy given that y(0)  0 . dx 5) Using Modified Euler Method, find the solution of initial value problem dy  log( x  y ), y(0)  2 at x  0.2 by assuming h  0.2 . dx dy 6) Using Modified Euler Method, find y(0.2), y(0.1) given  x 2  y 2 ; y(0)  1 . dx

form

 Runge – Kutta method for 1st and 2nd order equations 1) Find the value of y(1.1) using Runge – Kutta method of the fourth order given that dy  y 2  xy; y(1)  1 . dx 2) Using Runge – Kutta Method of fourth order solve

dy y 2  x 2  with y(0)  1 at dx y 2  x 2

x  0.2, 0.4 .

3) Using the Runge – Kutta method, tabulate the solution of the system dy dz  x  z,  x  y, y  0, z  1 when x  0 at intervals of h  0.1 from x  0.0 dx dx to x  0.2 . Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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d2 y dy  2  2 y  e 2 x sin x with 4) Consider the second order initial value problem 2 dx dx  y(0)  0.4 and y (0)  0.6 using fourth order Runge – Kutta find y(0.2) . 5) Given y  xy  y  0, y(0)  1, y(0)  0 . Find the value of y(0.1) by using RungeKutta method of fourth order. 2

d2 y  dy   x    y 2  0 using Runge – Kutta method for x  0.2 correct to 4 2 dx  dx  decimal places. Initial conditions are x  0, y  1, y  0 .

6) Solve

 Milne’s predictor – corrector methods for 1st order eqn. dy  y  x 2 is satisfied by y(0)  1, y(0.2)  1.12186, dx y(0.4)  1.46820, y(0.6)  1.7379 . Compute the value of y(0.8) by Milne’s predictor

1) The differential equation

– corrector formula. 2) Using Milne’s method find y(4.4) given 5 xy  y 2  2  0 given y(4)  1; y(4.1)  1.0049; y(4.2)  1.0097 and y(4.3)  1.0143 .

dy  1  x 3) Use Milne’s predictor – corrector formula to find y(0.4) , given  dx 2 y(0)  1, y(0.1)  1.06, y(0.2)  1.12 and y(0.3)  1.21 .

2

y

2

,

dy  xy  y 2 , y(0)  1 at x  0.1,0.2 and 0.3 dx continue the solution at x  0.4 by Milne’s predictor corrector method.

4) Using Taylors series method, solve

5) Using Runge – Kutta method of order 4 find y for x  0.1,0.2,0.3 given that dy  xy  y 2 , y(0)  1 . Continue the solution at x  0.4 using Milne’s method. dx

 Finite difference methods for solving 2nd order equation 1) Solve the differential equation

d2 y 1  y  x with y(0)  0, y(1)  0 with h  . 2 dx 4

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

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Engineering Mathematics

2015

d2 y  y  1  0, y(0)  y(1)  0 2) Solve the boundary value problem at x  0.5, dx 2 1 with h  . 4 3) Use finite difference method to find y(0.25), y(0.5) and y(0.75) satisfying the differential equation

d2 y  y  x subject to the conditions y(0)  0, y(1)  2 . dx 2

4) Solve the boundary value problem y  64 y  10  0 with y(0)  y(1)  0 by finite difference method. Find the value of y(0.5) .

---- All the Best ----

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917)

Page 20