## THIRD SEMESTER MATHEMATICS – PAPER – III

### (ALGEBRA II AND DIFFERENTIAL EQUATIONS I)

#### (4 lecture hours per week+3 hours of practical/week per batch of not morethan 15 students) (56 HOURS) THEORY

Group Theory (28 hrs) Definition and examples of groups – Some general properties of Groups, Group of permutations - cyclic permutations – Even and odd permutations. Powers of an element of a group – Subgroups – Cyclic groups problems and theorems. Cosets, Index of a group, Langrange’s theorem, consequences. Normal Subgroups, Quotient groups – Homomorphism – Isomorphism - Automorphism – Fundamental theorem of homomorphism

Differential Equations-I (28 hrs) Recapitulation of Definition, examples of differential equations, formation of differential equations by elimination of arbitrary constants, Differential equations of first order - separation of variables, homogeneous differential equations. Exact differential equations, reducible to exact, Linear differential equations. The general solution of a linear equation – Integrating factors found by inspection. The determination of integrating factors, Bernoulli’s equation. Ordinary Linear differential equations with constant coefficients – complementary function – particular integral – Inverse differential operators. Cauchy – Euler differential equations – Simultaneous differential equations (two variables with constant coefficients)

Books for References:
1. Daniel A Murray – Introductory Course to Differential equations

2. Earl David Rainville and Philip Edward Bedient – A short course in Differential equations, Prentice Hall College Div; 6th edition.

3. I. N. Herstien – Topics in Algebra.

4. Joseph Gallian – Contemporary Abstract Algebra, Narosa Publishing House, New Delhi, Fourth Edition.

5. G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra.

6. J B Fraleigh – A first course in Abstract Algebra.

7. Michael Artin – Algebra, 2nd ed. New Delhi, India: PHI Learning Pvt. Ltd., 2011.

8. Vashista, A First Course in Modern Algebra, 11th ed.: Krishna PrakasanMandir, 1980.

9. R Balakrishan and N.Ramabadran, A Textbook of Modern Algebra, 1st ed. New Delhi, India: Vikaspublishing house pvt.Ltd., 1991.

10. M D Raisinghania, Advanced Differential Equations, S Chand and Co. Pvt.Ltd., 2013.

11. F.Ayres, Schaum's outline of theory and problems of Differential Equations, 1st ed. USA McGraw-Hill, 2010.

12. S Narayanan and T K Manicavachogam Pillay, Differential Equations.: S V Publishers Private Ltd., 1981.

13. G F Simmons, Differential equation with Applications and historical notes, 2nd ed.: McGraw- Hill Publishing Company, Oct 1991.

#### PRACTICALS – III Mathematics practical with FOSS tools for computer programs (3 hours/ week per batch of not more than 15 students) LIST OF PROBLEMS

1. Verifying whether given operator is binary or not.

2. To find identity element of a group.

3. To find inverse element of a group.

4. Finding all possible subgroups of a finite group.

5. Examples to verify Lagrange’s theorem.

6. Illustrating homomorphism and isomorphism of groups.

7. Verification of Normality of a given subgroup.

8. Verifying Cayley’s theorem and isomorphism theorems.

9. Examples for finding left and right coset and finding the index of a group.

10. Solution of Differential equation using Scilab/Maxima and plotting the solution-I.

11. Solution of Differential equation using Scilab/Maxima and plotting the solution-II.

12. Solution of Differential equation using Scilab/Maxima and plotting the solution-III.

13. Solution of Differential equations using Scilab/Maxima and Plotting the solution-IV.

14. Note: The above list may be changed annually with the approval of the BOS in UG (Mathematics). Geogebra/Octave may also be used in place of scilab/maxima