### Mathematics(Hons.) Semester II

### Differential Equation I :

**Total Marks : 150
Theory : 75
Practical : 50
Internal Assessment : 25**

Differential Equation & Mathematical Models, order and degree of a differential equation, exact differential equations and integrating factors of first order differential equations, reducible second order differential equations, application of first order differential equations to acceleration-velocity model, growth and decay model.

**References :**

- [2]: Chapter 1 (Sections 1.1, 1.4, 1.6), Chapter 2 (Section 2.3) [3]: Chapter 2. Introduction to compartmental models, lake pollution model (with case study of Lake Burley Griffin), drug assimilation into the blood (case of a single cold pill, case of a course of cold pills, case study of alcohol in the bloodstream) , exponential growth of population, limited growth of harvesting.
- [1]: Chapter 2 (Sections 2.1,2.5-2.8), Chapter 3 (Sections 3.1-3.3) General solution of homogeneous equation of second order, principle of superposition for a homogeneous equation. Workskian, its properties and applications, Linear homogeneous and non-homogeneous equations of higher order with constant coefficients, Euler’s equation, method of undetermined coefficients, method of variation of parameters, application of second order differential equations to mechanical vibrations.
- [2]: Chapter 3 (Section 3.1-3.5) Equilibrium points, Interpretation of the phase plane , predator-prey model and its analysis, competing species and its analysis , epidemic model of influenza and its analysis, battle model and its analysis.
- 1]: Chapter 5 (Sections 5.1,5.3-5.4,5.6-5.7), Chapter 6

### Analysis II :

**Total Marks : 150
Theory : 75
Internal Assessment : 25**

Limits of functions, sequential criterion for limits, divergence criteria, review of limits theorems and one-sided limits, continuous functions, sequential criterion for continuity, discontinuity criterion, Dirichlet’s nowhere continuous functions, continuous functions on intervals, boundedness theorem, the maximum-minimum theorem, location of roots theorem, Bolzano’s intermediate value property, preservation of interval property.

**References :**

- [1]: Chapter 4 (Sections 4.1-4.3), Chapter 5 (Sections 5.1-5.3)
- [2]: Chapter 3 (Sections 17,18 and 20)

Uniform continuity, uniform continuity theorem, differentiation combinations of differentiable functions, Caratheodory theorem, chain rule, derivative of inverse functions, interior extremum theorem, intermediate value property for derivatives (Darboux’s theorem), review of Rolle’s theorem, Cauchy’s mean value theorem. - [1]: Chapter 5 (Section 5.4 upto 5.4.3), Chapter 6 (Sections 6.1-6.2,6.3.2)
- [2]: Chapter 3 (Section 19), Chapter 5 (Sections 28,29))

Taylor’s theorem with Lagrange and Cauchy form of remainders, binomial series theorem, Taylor series, Maclaurin series, expansionsof exponential, logarithmic and trigonometric functions, convex functions, applications of mean value theorems and Taylor’s theorem to monotone functions. Power Series, radius of convergence, interval of convergence. - [1]: Chapter 6 (Sections 6.4(upto 6.4.6)), Chapter 9 (Section 9.4 (page271)).
- R.G.Bartle and D.R.Sherbert , Introduction to Real analysis (3rd Edition), John Wiley and Sons (Asia) Pvt.Ltd., Singapore,2002.
- K.A.Ross, Elementary Analysis: The Theory of Calculus, Undergraduate Texts in Mathematics, Springer (SIE), Indian reprint, 2004.

### Probability And Statistics :

**Total Marks : 150
Theory : 75
Internal Assessment : 25**

Sample space, probability axioms, real random variables (discrete and continuous), cumulative distribution function, probability mass/density functions, mathematical expectation, moments, moment generating function , characteristic function, discrete distributions, uniform, binomial, Poisson, geometric, negative binomial, continuous distributions, uniform, normal, exponential .

**References :**

- [1]: Chapter 1 (Sections 1.1,1.3,1.5-1.9).
- [2]: Chapter 5 (Sections 5.1-5.5,5.7,chapter 6( Sections 6.2,6.3,6.5,6.6).

Joint cumulative distribution function and its properties, joint probability density functions, marginal and conditional distributions, expectation of function of two random variables, conditional expectations, independent random variables, bivriate normal distribution, correlation coefficient, joint moment generating function (mgf) and calculation of covariance , linear regression for two variables. - [1]: Chapter 2 (Sections 2.1, 2.3-2.5)
- [2]: Chapter 4 (Exercise 4.47), Chapter 6 (Section 6.7), Chapter 14 (Sections 14.1,14.2)

Chebyshev’s inequality, statement and interpretation of (weak) law of large numbers and strong law of large numbers , Central limit theorem for independent and identically distributed random variables with finite variance , Markov Chains, Chapman-Kolmogorov equations, classifications of states. - 2]: Chapter 4 (Section 4.4)
- [3]: Chapter 2 (Section 2.7), Chapter 4 (Sections 4.1-4.3)
- Robert V. Hogg, Joseph W. McKean and Allen T. Craig , Introduction to Mathematical Statistics, Pearson Education, Asia, 2006.
- Irwin Miller and Marylees Miller , John E Freund’s Mathematical Statistics with Applications (7th Edition) , Pearson Education, Asia, 2006.
- Sheldon Ross, Introduction to Probability Models (9th Edition), Academic Press, Indian Reprint, 2007.