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
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:
- Daniel A Murray – Introductory Course to Differential equations
- Earl David Rainville and Philip Edward Bedient – A short course in Differential
equations, Prentice Hall College Div; 6th edition.
- I. N. Herstien – Topics in Algebra.
- Joseph Gallian – Contemporary Abstract Algebra, Narosa Publishing House, New Delhi,
- G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra.
- J B Fraleigh – A first course in Abstract Algebra.
- Michael Artin – Algebra, 2nd ed. New Delhi, India: PHI Learning Pvt. Ltd., 2011.
- Vashista, A First Course in Modern Algebra, 11th ed.: Krishna PrakasanMandir, 1980.
- R Balakrishan and N.Ramabadran, A Textbook of Modern Algebra, 1st ed. New Delhi, India:
Vikaspublishing house pvt.Ltd., 1991.
- M D Raisinghania, Advanced Differential Equations, S Chand and Co. Pvt.Ltd., 2013.
- F.Ayres, Schaum's outline of theory and problems of Differential Equations, 1st ed. USA
- S Narayanan and T K Manicavachogam Pillay, Differential Equations.: S V Publishers Private
- 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
- Verifying whether given operator is binary or not.
- To find identity element of a group.
- To find inverse element of a group.
- Finding all possible subgroups of a finite group.
- Examples to verify Lagrange’s theorem.
- Illustrating homomorphism and isomorphism of groups.
- Verification of Normality of a given subgroup.
- Verifying Cayley’s theorem and isomorphism theorems.
- Examples for finding left and right coset and finding the index of a group.
- Solution of Differential equation using Scilab/Maxima and plotting the solution-I.
- Solution of Differential equation using Scilab/Maxima and plotting the solution-II.
- Solution of Differential equation using Scilab/Maxima and plotting the solution-III.
- Solution of Differential equations using Scilab/Maxima and Plotting the solution-IV.
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