JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Prerequisites

Foundation course (No prerequisites).

Course Objectives:

To learn

• types of matrices and their properties
• the concept of rank of a matrix and applying the same to understand the consistency
• solving the linear systems
• the concepts of eigen values and eigen vectors and reducing the quadratic forms into their canonical forms
• partial differentiation, concept of total derivative
• finding maxima and minima of functions of two variables
• methods of solving the linear differential equations of first and higher order
• the applications of the differential equations
• formation of the partial differential equations and solving the first order equations.

Course Outcomes:

After learning the contents of this paper the student must be able to

• write the matrix representation of a set of linear equations and to analyze the solution of the system of equations
• find the Eigen values and Eigen vectors which come across under linear transformations
• find the extreme values of functions of two variables with/ without constraints.
• identify whether the given first order DE is exact or not
• solve higher order DE’s and apply them for solving some real world problems

UNIT - I

Initial Value Problems and Applications Exact differential equations - Reducible to exact. Linear differential equations of higher order with constant coefficients: Non homogeneous terms with RHS term of the type e ax , sin ax, cos ax, polynomials in x, e ax V(x), xV(x)- Operator form of the differential equation, finding particular integral using inverse operator, Wronskian of functions, method of variation of parameters. Applications: Newton’s law of cooling, law of natural growth and decay, orthogonal trajectories, Electrical circuits.

UNIT - II

Linear Systems of Equations Types of real matrices and complex matrices, rank, echelon form, normal form, consistency and solution of linear systems (homogeneous and Non-homogeneous) - Gauss elimination, Gauss Jordon and LU decomposition methods- Applications: Finding current in the electrical circuits.

UNIT - III

Eigen values, Eigen Vectors and Quadratic Forms Eigen values, Eigen vectors and their properties, Cayley - Hamilton theorem (without proof), Inverse and powers of a matrix using Cayley - Hamilton theorem, Diagonalization, Quadratic forms, Reduction of Quadratic forms into their canonical form, rank and nature of the Quadratic forms – Index and signature.

UNIT - IV

Partial Differentiation Introduction of partial differentiation, homogeneous function, Euler’s theorem, total derivative, Chain rule, Taylor’s and Mclaurin’s series expansion of functions of two variables, functional dependence, Jacobian. Applications: maxima and minima of functions of two variables without constraints and Lagrange’s method (with constraints)

UNIT - V

First Order Partial Differential Equations Formation of partial differential equations by elimination of arbitrary constants and arbitrary functions, Lagranges method to solve the first order linear equations and the standard type methods to solve the non linear equations.

Text Books:

• 1. A first course in differential equations with modeling applications by Dennis G. Zill, Cengage Learning publishers.
• 2. Higher Engineering Mathematics by Dr. B. S. Grewal, Khanna Publishers.

References:

• 1. Advanced Engineering Mathematics by E. Kreyszig, John Wiley and Sons Publisher.
• 2. Engineering Mathematics by N. P. Bali, Lakshmi Publications.