1

NORTH MAHARASHTRA UNIVERSITY JALGAON.

Syllabus for S.Y.B.Sc. (Mathematics)

With effect from June 2013. (Semester system).

The pattern of examination of theory papers is semester system. Each theory course is of

50 marks (40 marks external and 10 marks internal) and practical course is of 100 marks

(80 marks external and 20 marks internal). The examination of theory courses will be

conducted at the end of each semester and examination of practical course will be

conducted at the end of the academic year.

STRUCTURE OF COURSES

SEMESTER –I

MTH-231 : Advanced Calculus

MTH-232 (A) : Topics in Algebra

OR

MTH-232 (B) : Computational Algebra

SEMESTER- II.

MTH-241 : Complex Analysis

MTH-242 (A) : Topics in Differential Equations

OR

MTH-242(B) : Differential Equations and Numerical methods

MTH-203 : Practical Course based on

MTH-231, MTH-232, MTH-241, MTH-242

…………………………………………………………………………………………………

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SEMESTER – I

MTH -231: Advanced Calculus

Unit- 1 : Functions of two and three variables Periods -15 ,Marks – 10

1.1 Limits and Continuity

1.2 Partial Derivatives and Jacobians .

1.3 Higher order Partial Derivatives

1.4 Differentiability and Differentials

1.5 Necessary and sufficient conditions for Differentiability

1.6 Schwarz’s Theorem

1.7 Young’s Theorem.

Unit- 2 : Composite Functions and Mean Value Theorem Periods – 15, Marks – 10

2.1 Composite functions. Chain Rules.

2.2 Homogeneous functions.

2.3 Euler’s Theorem on Homogeneous Functions.

2.4 Mean Value Theorem for Function of Two Variables.

Unit -3 : Taylor’s Theorem and Extreme Values Periods – 15, Marks – 10

3.1 Taylor’s Theorem for a function of two variables.

3.2 Maclaurin’s Theorem for a function of two variables.

3.3 Absolute and Relative Maxima and Minima.

3.4 Necessary Condition for extrema.

3.5 Critical Point, Saddle Point.

3.6 Sufficient Condition for extrema.

3.7 Lagrange’s Method of undetermined multipliers.

Unit -4 : Double and Triple Integrals Periods – 15, Marks – 10

4.1 Double integrals by using Cartesian and Polar coordinates.

4.2 Change of Order of Integration.

4.3 Area by Double Integral.

4.4 Evaluation of Triple Integral as Repeated Integral.

4.5 Volume by Triple Integral.

Reference Books –

1. Mathematical Analysis: S.C. Malik and Savita Arora. Wiley Eastern Ltd, New Delhi.

2. Calculus of Several Variables: Schaum Series.

3. Mathematical Analysis: T.M.Apostol. Narosa Publishing House, New Delhi, 1985

4. A Course of Mathematical Analysis: Shanti Narayan, S. Chand and Company, New Delhi.

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MTH -232(A): Topics in Algebra

Unit-1 : Groups Periods-15, Marks-10.

1.1 Definition of a group.

1.2 Simple properties of group.

1.3 Abelian group.

1.4 Finite and infinite groups.

1.5 Order of a group.

1.6 Order of an element and its properties.

Unit-2 : Subgroups Periods-15, Marks-10.

2.1 Subgroups. Definition, criteria for a subset to be a subgroup.

2.2 Cyclic groups.

2.3 Coset decomposition.

2.4 Lagrange’s theorem for finite group.

2.5 Euler’s theorem and Fermat’s theorem.

Unit -3 : Homomorphism and Isomorphism of Groups Periods-15, Marks-10

3.1 Definition of Group Homomorphism and its Properties .

3.2 Kernel of Homomorphism and Properties.

3.3 Definition of Isomorphism and Automorphism of Groups.

3.4 Properties of Isomorphism of Groups.

Unit-4 : Rings Periods-15, Marks-10

4.1 Definition and Simple Properties of a Ring.

4.2 Commutative Ring, Ring with unity, Boolean Ring.

4.3 Ring with zero divisors and without zero Divisors.

4.4 Integral Domain, Division Ring and Field. Simple Properties.

Reference Books –

1. Topics in Algebra : I. N. Herstein.

2. A first Course in Abstract Algebra : J. B. Fraleigh.

3. University Algebra : N. S. Gopalkrishanan.

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MTH -232(B): Computational Algebra

Unit-1 : Groups Periods-15, Marks-10.

1.1 Definition of a group.

1.2 Simple properties of group.

1.3 Abelian group.

1.4 Finite and infinite groups.

1.5 Order of a group.

1.6 Order of an element and its properties.

Unit-2 : Subgroups Periods-15, Marks-10.

2.1 Subgroups. Definition, criteria for a subset to be a subgroup.

2.2 Cyclic groups.

2.3 Coset decomposition.

2.4 Lagrange’s theorem for finite group.

2.5 Euler’s theorem and Fermat’s theorem.

Unit -3 : Homomorphism and Isomorphism of Groups Periods-15, Marks-10

3.1 Definition of group homomorphism and its properties .

3.2 Kernel of homomorphism and properties.

3.3 Definition of isomorphism and automorphism of groups.

3.4 Properties of isomorphism of groups.

Unit-4 : Group Codes Periods -15, Marks 10

4.1 Message, Word, Encoding Function, Code Word.

4.2 Detection of K or Fewer Error, Weight Parity, Check Code

4.3 The Hamming Distance, Properties of the Distance Function, The Minimum Distance of an

Encoding function.

4.4 Group Codes.

4.5 (n, m) Decoding function, Maximum Likelihood Decoding Function.

4.6 Decoding Procedure for a Group Code Given by a Parity Matrix.

Reference Books –

1. Discrete Mathematical Structure : Bernard Kolman Robert C. Busby, Ross Prentice Hall and India

New Delhi. Eastern Economy Edition.

2. Topics in Algebra : I.N. Herstein.

3. A first Course in Abstract Algebra : J. B. Fraleigh.

4. University Algebra : N. S. Gopalkrishanan.

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SEMESTER – II

MTH -241 : Complex Analysis

Unit-1 : Complex numbers Period-15, Marks-10

1.1 Algebra of complex numbers

1.2 Representation of complex numbers as an order pair

1.3 Modulus and amplitude of a complex number

1.4 Triangle inequality and Argand’s diagram

1.5 De Moivre’s theorem for rational indices

1.6 nth

roots of a complex number

Unit-2: Functions of complex variables Period-15, Marks-10

2.1 Limits, Continuity, Derivative.

2.2 Analytic functions, Necessary and sufficient condition for analytic function.

2.3 Cauchy Riemann equations.

2.4 Laplace equations and Harmonic functions

2.5 Construction of analytic functions

Unit-3 : Complex integrations Period-15, Marks-10

3.1 Line integral and theorems on it.

3.2 Statement and verification of Cauchy-Gaursat’s Theorem.

3.3 Cauchy’s integral formulae for f(a) and f’(a)

3.4 Taylor’s and Laurent’s series.

Unit-4 : Calculus of Residues Period-15, Marks-10

4.1 Zeros and poles of a function.

4.2 Residue of a function

4.3 Cauchy’s residue theorem

4.4 Evaluation of integrals by using Cauchy’s residue theorem

4.5 Contour integrations of the type

Reference books –

1. Complex Variables and Applications ; R.V.Churchill and J.W. Brown

2. Theory of Functions of Complex Variables : Shanti Narayan S. Chand and Company, New Delhi.

3. Complex variables : Schaum Series.

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MTH-242(A): Topics in Differential Equations

Unit-1 : Theory of ordinary differential equations Period-15, Marks-10

1.1 Lipschitz condition

1.2 Existence and uniqueness theorem

1.3 Linearly dependent and independent solutions

1.4 Wronskian definition

1.5 Linear combination of solutions

1.6 Theorems on i) Linear combination of solutions

ii) Linearly independent solutions

iii) Wronskian is zero

iv) Wronskian is non zero

1.7 Method of variation of parameters for second order L.D.E.

Unit-2 : Simultaneous Differential Equations Period-15, Marks-10

2.1 Simultaneous linear differential equations of first order

2.2 Simultaneous D.E. of the form

2.3 Rule I: Method of combinations

2.4 Rule II: Method of multipliers

2.5 Rule III: Properties of ratios

2.6 Rule IV: Miscellaneous

Unit-3 : Total Differential or Pfaffian Differential Equations Period-15, Marks-10

3.1 Pfaffian differential equations

3.2 Necessary and sufficient condition for integrability

3.3 Conditions for exactness

3.4 Method of solution by inspection

3.5 Solution of homogenous equation

3.6 Use of auxiliary equations

Unit-4 : Beta and Gamma Functions Period-15, Marks-10

4.1 Introduction

4.2 Euler’s Integrals: Beta and Gamma functions

4.3 Properties of Gamma function

4.4 Transformation of Gamma function

4.5 Properties of Beta function

4.6 Transformation of Beta function

4.7 Duplication formula

Reference Book-

1. Ordinary and Partial Differential Equation by M. D. Rai Singhania, S. Chand & Co.(11th

Revised

Edition)

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MTH-242(B): Differential Equations and Numerical methods

Unit-1 : Theory of ordinary differential equations Period-15, Marks-10

1.1 Lipschitz condition

1.2 Existence and uniqueness theorem

1.3 Linearly dependent and independent solutions

1.4 Wronskian definition

1.5 Linear combination of solutions

1.6 Theorems on i) Linear combination of solutions

ii) Linearly independent solutions

iii) Wronskian is zero

iv) Wronskian is non zero

1.7 Method of variation of parameters for second order L.D.E.

Unit-2 : Simultaneous Differential Equations Period-15, Marks-10

2.1 Simultaneous linear differential equations of first order

2.2 Simultaneous D.E. of the form

2.3 Rule I: Method of combinations

2.4 Rule II: Method of multipliers

2.5 Rule III: Properties of ratios

2.6 Rule IV: Miscellaneous

Unit-3 : Total Differential or Pfaffian Differential Equations Period-15, Marks-10

3.1 Pfaffian differential equations

3.2 Necessary and sufficient condition for integrability

3.3 Conditions for exactness

3.4 Method of solution by inspection

3.5 Solution of homogenous equation

3.6 Use of auxiliary equations

Unit-4 : Numerical Methods for solving ODE Period-15, Marks-10

4.1 Picard’s method

4.2 Taylor series method

4.3 Modified Euler’s method

4.4 Runge-Kutta method of fourth order

4.5 Milne’s Method

4.6 Adam Bashforth method

Reference Book-

1. Ordinary and Partial Differential Equation by M.D.Rai Singhania, S.Chand & Co.(11th

Revised

Edition)

2. Numerical Analysis By S.S. Sastry

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NORTH MAHARASHTRA UNIVERSITY JALGAON

S.Y.B.Sc. Mathematics

Practical Course MTH-203

Based on MTH-231, MTH-232, MTH-241, MTH-242.

(With effect from June 2013)

Index

Practical

No.

Title of Practical

1 Functions of two and three variables

2 Composite Functions and Mean Value Theorem.

3 Taylor’s Theorem and Extreme Values.

4 Double and Triple Integrals

5 Groups

6 Subgroups

7 Homomorphism and Isomorphism of Groups

8(A) Rings

8(B) Group Codes

9 Complex Numbers

10 Functions of complex variables

11 Complex Integration

12 Calculus of Residues

13 Theory of Ordinary Differential Equations

14 Simultaneous Differential Equations

15 Total Differential or Pfaffian Differential Equations

16(A) Beta and Gamma functions

16(B) Numerical Methods for solving Ordinary Differential Equations

9

Practical No. 1 : Functions of two and three variables

1. a) Evaluate

along the path y = x

2

b) Prove that

does not exists.

2. a)

b) Show that the function

3. a) If

b)

4. Let

5. a)

b)

…………………………………………………………………………………………………

10

Practical No. 2 :

Composite Functions and Mean Value Theorem.

1. a)

b) Show that

2. a) If

.

b) If

3. a) If

.

b) If

.

4 a) If

b)

.

5 If then show that used in mean value theorem applied to the points (2 , 1) and

(4 , 1) satisfies the quadratic equation,

…………………………………………………………………………………………………

Practical No. 3 : Taylor’s Theorem and Extreme Values. 1. a) Expand in the powers of using Taylor’s expansion.

b) Expand about (1 , 1) up to third degree terms using Taylor’s expansion

2. a) Show that

b) Expand in the powers of x and y up to third degree terms

3. Expand

upto the third degree terms and hence find

4 a) Find the maxima and minima of

b) Investigate the maxima and minima of

5 a) Classify the critical points of

b) Find the shortest distance of from the origin.

…………………………………………………………………………………………………

11

Practical No. 4 : Double and Triple Integrals

1. Evaluate

2 2

2 2 2

0 0

a ya

a x y dydx

2. Evaluate 2 2( )x y dxdy , over the region in the positive quadrant for which

3. Change the order of the integration 2

2 3

04

( , )

a a x

xa

f x y dxdy

4. a) Evaluate 2

0 0

x

x yxe dxdy

b) Find the area of the ellipse

5. a) Evaluate ( 2 )R

x y z dxdydz , where R :

b) Find the volume of the sphere by using triple integral.

…………………………………………………………………………………………………

Practical No. 5 : Groups

1. a) Show that the set of all positive rational numbers forms an abelian group under the binary operation

defined by

b) Show that G = -{-1} is an abelian group under the binary operation

, a, b G.

2. a) Let G = { (a, b) : a, b , a ≠ 0 }. Show that (G, ) is a non-abelian group,

where (a, b) (c, d) = (ac, bc+d).

b) Prove that

is a group under matrix multiplication.

3. a) In ( , ), find i) ii) iii) iv) 0( v) 0(

b) In ( , ), find i) ii) iii) iv) 0( v) 0(

4. If G is a group such that a2 = e, for all a, bG, then show that G is abelian.

5. If in a group G, a5 = e and aba

– 1 = b

2 for a, bG, then find order of b.

…………………………………………………………………………………………………

Practical No. 6 : Subgroups

1. If H is a subgroup of a group G and xG, then show that xHx-1

= { xhx-1

: hH } is a subgroup of G.

2. a) Let G be a group of all nonzero complex number under multiplication.

Show that H = { a + ib G : a2 + b

2 = 1 } is subgroup of G.

b) Determine whether and are subgroups of 12.

3. a) Show that is a cyclic group. Find all its generators, all its proper subgroups and the

order of every element.

b) Find all subgroups of .

4. a) Let . Find all subgroups of G.

b) Let G = 6 and be subgroup of G. Find all right and left cosets of H.

5. a) Show that every proper subgroup of a group of order 77 is cyclic.

b) Find the remainder obtained when 3319

is divisible by 7.

…………………………………………………………………………………………………

12

Practical No. 7 : Homomorphism and Isomorphism of Groups

1. Prove that the mapping f : → 0 such that f(z) = ez is a homomorphism of the additive group of

complex numbers onto the multiplicative group of nonzero complex number. What is the kernel of f.

2. Let G = {(a, b) : a, b } be a group under addition defined by (a, b) + (c, d) = (a + c, b + d) .

Show that the mapping f : G → defined by f(a, b) = a, (a, b)G, is a homomorphism and find its

kernel.

3. a) Let G be a group of all matrices of the type

under matrix

multiplication and G’ be a group of nonzero complex numbers under multiplication. Show that f : G →

G’ defined by

is an isomorphism.

b) If G = { 1, -1, i, -i } is a group under multiplication and is a group under

multiplication modulo 10, then show that G and are isomorphic.

4. a) Let G be a group and G. Show that f : G → G defined by f(a) = xax−1

, aG is an automorphism.

b) Let * be the multiplicative group of nonzero complex numbers and * be the multiplicative group

of nonzero real numbers. Define : * → * by for z *. Show that is a

homomorphism. Is an isomorphism ? Explain.

5. Let G be a group and f : G → G be a map defined by f(x) = x-1, G. Prove that

a) If G is abelian, then f is an isomorphism.

b) If f is a group homomorphism, then G is abelian.

…………………………………………………………………………………………………

Practical No. 8(A) : Rings

1. a) Let be the set of even integers and is a binary operation on is defined as

, where

ab is ordinary multiplication. Show that ( , +, ) is a commutative ring, where + is ordinary

addition in .

b) For a, b define a b = a + b −1 and a b = a + b − ab . Show that ( , , ) is

commutative ring with identity element.

2. Show that = {a + ib : a, b }, the set of Gaussian integers, forms an integral domain under

usual addition and multiplication of complex numbers.

3. Show that is an integral domain.

4. If R is a ring in which x2 = x, for every x R, then prove that

i) x + x = 0 ii) x + y = 0 x = y iii) R is commutative.

5. a) In the ring , find

i) the zero element of the ring ii) units in iii) additive inverse of 3

iv) multiplicative inverse of 3 v) -8 3 vi) 4 (-8)

b) Which of the following are fields :

i) ii) iii) iv) v) . Justify your answer.

…………………………………………………………………………………………………

13

Practical No. 8(B) : Group Codes

1. a) Consider the encoding function defined by

, , ,

, , ,

,

i. Find the minimum distance of e.

ii. How many errors will e detect?

b) Show that the encoding function defined by

, , ,

, , ,

, is a group code. Also find the minimum distance of e.

2. Compute:

i.

ii.

3. Let

be a parity check matrix . Determine the group code .

4. Consider the parity check matrix :

Determine the coset leaders for . Also compute the Syndrome for each coset leader and

decode the code relative to maximum likelihood decoding function.

5. Let the decoding function defined by , where

has atleast two 1’s.

has less than two 1’s,

, then determine , where

i.

ii.

…………………………………………………………………………………………………

14

Practical No.9 : Complex Numbers

1. Find the modulus argument form (polar form) of following complex numbers :

a) 3 + 4i b) - i

2. If represent the vertices of an equilateral triangle, then prove that

3. a) If , then prove that

b) Prove that

= 1 , if either

4. If cos + cos + cos = 0 and sin + sin + sin = 0 , then show that

a) cos3 + cos3 + cos3 = 3 cos( + + ) and sin3 + sin3 + sin3 = 3 sin( + + )

b) cos2 + cos2 + cos2 = 0 = sin2 + sin2 + sin2

5. a) Find all values of

b) Solve the equation x4 – x

3 + x

2 – x + 1 = 0

…………………………………………………………………………………………….

Practical No. 10 : Functions of complex variables

1. a) Evaluate

.

b) Show that

does not exist.

2. Discuss the continuity of

,

= , if

at

3. Construct an analytic function and express it in terms of

.

4. Prove that is harmonic and find its harmonic conjugate. Also find the

corresponding analytic function.

5. is analytic, then prove that

.

…………………………………………………………………………………………………………..

15

Practical No. 11 : Complex Integration

1. Expand

in powers of for

b) c)

2. Expand

in Laurent’s series valid for

a) b)

3. Evaluate

where C is

a) The Straight line joining

b) The straight line joining first and then

4. Verify Cauchy’s theorem for taken round the unit circle

5. Use Cauchy’s integral formula to evaluate:

a)

b)

…………………………………………………………………………………………………………..

Practical No. 12 : Calculus of Residues

1. Compute the residue at double poles of

.

2. Evaluate

by Cauchy’s residue theorem, where C is

a) The Circle b) The Circle

3. Evaluate by Contour integration :

.

4. Apply Calculus of residues to evaluate

.

5. Use Calculus of residues to evaluate

…………………………………………………………………………………………………………..

16

Practical No. 13 : Theory of Ordinary Differential Equations

1. Show that y = 3e2x

+ e-2x

- 3x is the unique solution of the initial value problem 4 12 ,y y x where

(0) 4, (0) 1.y y

2. Find the unique solution of 1y where (0) 1y and (0) 2.y

3. Show that f(x,y) = xy2 satisfies the Lipschitz condition on the rectangle R: 1, 1x y but does not

satisfy the Lipschitz condition on the strip S : 1,x y .

4. Prove that sin2x and cos2x are solutions of 4 0y y and these solutions are linearly independent.

5. a) Show that exsinx and e

xcosx are linearly independent solutions of the differential equation

2 2 0y y y . What is the general solution? Find the solution y(x) if y(0) = 2 and (0) 0y .

b) Prove that 1, x, x2 are linearly independent. Hence, form the differential equation whose solutions

are 1, x and x2.

……………………………………………………………………………………

Practical No. 14 : Simultaneous Differential Equations

1. a) Solve 1, 1,dx dy

y xdt dt

by the method of differentiation.

b) Solve 25 , 3 ,t tdx dyx y e x y e

dt dt by operator method.

2. a) Solve 2 2 2 2 2

dx dy dz

y x x y z b) Solve

2 22

dx dy dz

y x xy xz

3. a) Solve ( ) ( ) ( )

adx bdy cdz

b c yz c a xz a b yx

b) Solve

2 2 2 2 2 2( ) ( ) ( )

dx dy dz

y z x z x y x y z

4. Solve 2 2 2( ) ( ) ( )

dx dy dz

y x y z y z x z x

5. a) Solve 2 2 2 2 2

dx dy dz

x y z xy xz

b) Solve

dx dy dz

y z z x x y

……………………………………………………………………………………

17

Practical No. 15 : Total Differential or Pfaffian Differential Equations

1. Show that the following differential equation is integrable and hence solve it:

a) 2 2 2( ) 2 2 0y z x dx xydy xzdz

b) 2 2 2 2 2 2( ) ( ) ( ) 0yz x yz dx zx y zx dy xy z xy dz

2. Solve the following by the method of homogenous differential equations :

a) 2( ) ( ) (2 ) 0y z dx x y dy y x z dz

b) 2 2 2( ) ( ) ( ) 0y yz dx z xz dy y xy dz

3. Solve the following by the method of homogenous differential equations :

a) 2 2(2 ) (2 ) ( ) 0xz yz dx yz xz dy x xy y dz

b) 2 2 2( 2 ) (2 ) 0z dx z yz dy y yz xz dz

4. Solve the following by the method of auxiliary equations :

a) 3 2 0xz dx zdy yzdz

b) 2 2(2 ) (2 ) ( ) 0xz yz dx yz zx dy x xy y dz

c) 2 2 2 2 2 2( ) ( ) ( ) 0y yz z dx z xz x dy x yx y dz

……………………………………………………………………………………

Practical No. 16(A) : Beta and Gamma functions

1. a) Evaluate 6 2

0

xx e dx

b) Prove that 9 32

2 945

2. a) Prove that 1

2 1.3.5.....(2 1)2

n n n

b) Show that 2 2 1

0 2

ax n

n

ne x dx

a

( n>0 )

3. a) Show that

10

1( 0)

(log )

a

x a

axdx a

a a

b) Show that

2 22

0 2

ax x ae dx e

4. a) Show that

12

3 3

0

16(8 )

2 3x x dx

b) Evaluate

21

0 2

x dx

x

5. Evaluate 1

60 1

x dx

x

……………………………………………………………………………………

18

Practical No. 16(B) :

Numerical Methods for solving Ordinary Differential Equations

1. a) Using Picard’s method of successive approximations obtain a solution upto third approximation

of the differential equation 2dy

x ydx

with y(0) = 0.

b) Using Picard’s method of successive approximations obtain a solution upto third approximation

of the differential equation 4dy

x y xdx

with y(0) = 3.

2. a) From the Taylor’s series for y(x), find y(0.1) correct upto four decimal places if y(x) satisfies

2dy

x ydx

and y(0) = 1.

b) From the Taylor’s series for y(x), find y(0.1) correct upto four decimal places if y(x) satisfies

2 3 xdyy e

dx and y(0) = 1 and compare with the exact equation.

3. a) Given 10log ( ),dy

x ydx

with the initial condition that y = 1 when x = 0, find y for x = 0.2,

using modified Euler’s method.

b) Given ,dy

x ydx

with the initial condition that y = 1 when x = 0, find y for x = 0.2,

using modified Euler’s method.

4. a) Using Runge-Kutta method of fourth order solve

2 2

2 2

dy y x

dx y x

with y(0) = 1 at x = 0.2 .

b) Find y(2) if y(x) is the solution of 2

dy y x

dx

, assuming y(0) = 2, y(0.5) = 2.636, y(1) = 3.595

and y(1.5)=4.968, using Milne’s method.

5. a) Find y(0.4) if y(x) is the solution of 21

dyxy

dx , assuming y(0) = 1, y(0.1) = 1.105,

y(0.2) = 1.223 and y(0.3) = 1.354, using Milne’s method.

b) Find y(1.4) if y(x) is the solution of 2(1 )

dyx y

dx , assuming y(1) = 1, y(1.1) = 1.233,

y(1.2) = 1.548 and y(1.3) = 1.979, using Adam-Bashforth method.

……………………………………………………………………………………