Module 1 (14 hours)
Matrix Theory: Gauss elimination method, Gauss-Jordon method for finding inverse of a matrix, Vector space, subspace, linear span, linear dependence and independence, Basis and dimension of vector space, Row and column spaces, Rank and nullity of a matrix, Rank and Nullity Theorem, Inner product spaces, Gram-Schmidt Orthogonalization, Matrix representation of Linear Transformations, Solvability of systems of linear equations, Eigen values, Eigenvectors, Diagonalization of matrices, Reduction of a quadratic form to canonical form.

Module 2 (12 hours)
ODE of the first order: Geometrical interpretations, Separable equations, Reduction to separable form, Exact equations, Integrating factors, Linear equations, Bernoulli equations, orthogonal trajectories, Existence and uniqueness of IVP (Picard’s Theorem), Applications to physical problems.

ODE of higher order: Fundamental system and general solutions of homogeneous equations of order two, Wronskian, reduction of order, Solution of non-homogeneous equations by method of undetermined coefficients and variation of parameters. Extension to higher order differential equations, Euler-Cauchy equation, Power series method, Applications to physical problems.

Module 3 (10 hours)
Laplace Transform: Laplace and inverse Laplace transforms, the existence of Laplace transform, first shifting theorem, transforms of derivative and integral, second shifting theorem, differentiation and integration of transforms, Convolution theorem, Solution of ordinary differential equations by using Laplace transform.

To introduce the concepts of vector spaces, basis, linear transformation, and inner product spaces.

To introduce the rank of a matrix and its application for testing the invertibility of a square matrix and the solvability of a linear system of equations.

To teach different methods of solving ordinary differential equations of one/higher order.

To introduce Laplace transform, its properties and applications in solving differential equations

CO1: Students will learn to solve systems of linear equations and find the inverse of a matrix using the Gauss-Jordan elimination method. Additionally, they will determine bases of finite-dimensional vector spaces and understand the significance of the Rank-Nullity Theorem.

CO2: Students will be able to compute eigenvalues and eigenvectors of a matrix and understand matrix diagonalization. They will also learn about inner products and how to transform a set of linearly independent vectors into an orthonormal set.

CO3: Students will become proficient in solving first-order ordinary differential equations and determining orthogonal trajectories for a given one-parameter family of curves. They will also study the fundamental system and general solutions of higher-order homogeneous equations, the Wronskian, and the Euler-Cauchy equation.

CO4: Students will learn to solve nonhomogeneous linear differential equations using the method of undetermined coefficients and the variation of parameters. Additionally, they will be introduced to the power series method.

CO5: Students will explore various techniques for determining the Laplace transform of functions and learn how to solve differential equations using the Laplace transform.

E. Kreyszig, Advanced engineering Mathematics, John Wiley & Sons, Inc.

G. F. Simmons and S. G. Krantz, Differential Equations: Theory, Technique and Practice, Tata McGraw-Hill

G. Strang, Linear Algebra and its applications, Cengage Learning

S. L. Ross, Differential Equations, John Wiley & Sons, Inc.