Differential Equations and Linear Algebra

Course Code (Credit):

CUTM1001 (2-0-1)

Course Objectives:

  • Introduce students to how to solve linear Differential Equations with different methods.
  • To solve the system of linear equations appearing in the problems of engineering domains.
  • To use Eigen values and Eigen vectors in Control theory, vibration analysis, circuits, dynamics, etc.
  • Introduce students how to solve first and second order differential equations.

Learning Outcomes:

  • Understand the importance of linear functions in mathematics.
  • Solve systems of linear equations using Gauss-elimination.
  • Learn concepts of ODEs and their applications in scientific and engineering problems.
  • Be competent in solving 1st & higher order ODEs analytically.

Course Syllabus:

Module I:

First order linear differential equations and its applications.

Project 1: Applications in RL-RC electrical circuits

Module II:

Second order linear homogeneous differential equations and its applications.

Project 2: RLC Circuit, Pendulum

Module III:

Non-homogeneous second-order differential equations. Finding particular integral using inverse operator method.

Project 3: Mass-spring system, Damped vibration

Module IV:

Matrices, Gauss elimination, linear dependence, rank.

Project 4: Traffic flow in one-way street networks

Module V:

Determinants and Cramer’s Rule, Linear system theorem

Module VI:

Eigenvalues and Eigenvectors.

Project 5: Markov model, Leslie model

Module VII:

Symmetric, Skew-Symmetric, Orthogonal Matrices

Project 6: Properties and implications of orthogonal matrices (rotation)

Text Books:

  1. Advanced Engineering Mathematics by Erwin Kreyszig, 8th Edition
  2. Higher Engineering Mathematics by B.V. Ramana

Reference Books:

  1. J. Sinha Roy & S. Padhy, A Course of Ordinary and Partial Differential Equations
  2. Thomas' Calculus – G.B. Thomas et al.
  3. Introduction to Real Analysis – R.G. Bartle & D.R. Sherbert

Session Plan:

Session Topic Reference Link (if any)
1Overview of differential equations-
2First order linear differential equations-
3Applications of First order differential equations (Kirchhoff's law)-
4-5Project-1: RL-RC electrical circuits-
62nd order homogeneous differential equations-
7Solution of 2nd order equations-
8Applications of second-order DEs-
9-10Project-2: RLC Circuit-
112nd order non-homogeneous DEs-
12Particular integral (Exponential)Video
13Particular integral (Trigonometric)Video
14-15Project-3: Spring systemsVideo
16Basic concepts of matricesPDF
17Gauss eliminationVideo
18Linearly independent vectors, Rank PDF
Video 1
Video 2
19-20Project-4: Traffic flow in one-way streets-
21Determinants and Cramer’s RuleVideo
22Existence & Uniqueness of solutions Slides
Video
23Eigenvalues and Eigen VectorsVideo
24Properties of Eigenvalues/VectorsVideo
25Applications of Eigenvalues/VectorsVideo
26-27Project-5: Markov & Leslie models-
28Symmetric, Skew-Symmetric, Orthogonal MatricesVideo
29Properties of these matricesVideo
30Problems on these matricesVideo
31-32Project-6: Orthogonal matrices and rotation-