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• Introduce students to how to solve linear Differential Equations with different methods.

• To solve the system of linear equations appearing in the problems of electrical engineering, mechanical engineering etc.

• To use Eigen values and Eigen vectors in Control theory, vibration analysis, electric circuits, advanced dynamics problems.

• Introduce students how to solve first order and second order differential equations

**Upon successful completion of this course, students will be able to:**

• Understand the importance of linear functions in mathematics.

• Solve systems of linear equations using Gauss- elimination to reduce to echelon form.

• Learn fundamental concepts of ODE theories and where and how such equations arise in applications to scientific and engineering problems.

• Be competent in solving linear/non-linear 1st & higher order ODEs using analytical methods to obtain their exact solutions

**Module-I:**

First order linear differential equations and its applications

**Project-1:** Some applications of differential equations in RL-RC electrical circuit problems

**Module-II:**

Second order linear homogeneous differential equations (Real roots, Real equal roots, Complex conjugate roots) and its applications.

**Project-2:** RLC Circuit, Pendulum

**Module-III:**

Second order linear non-homogeneous differential equations, Finding particular integral consisting of exponential, trigonometric (Sine, cosine) using inverse operator method

**Project-3:** Simple mass-spring system, Damped vibration system

**Module-IV:**

Basic concepts of a matrices, solution of linear system of equations by Gauss elimination method, linearly independent and dependent of a vectors, rank of a matrix.

**Project 4:** Report on finding the traffic flow in the net of one-way streets

**Module-V:**

Determinants and Cramer’s Rule, Fundamental theorem of linear system of equations.

**Module-VI:**

Eigenvalues and Eigen vectors of a matrix.

**Project 5:** (i)Find the limit states of the Markov process model.

(ii)Find the growth rate in the Leslie model

**Module-VII:**

Symmetric, Skew-Symmetric , Orthogonal Matrices and Properties

**Project 6:** To make a report to show that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations?

**Text Books**:

1. Advanced engineering mathematics by Erwin Kreyszig, 8th edition

Chapter-6 (6.1-6.6), Chapter-7 (7.1,7.3)

2. Higher Engineering by B.V. Ramana

Chapter-8(8.1,8.2,8.9,8.10,8.21), Chapter-9 (9.2,9.3,9.5)

**Reference Books**:

1. J. Sinha Roy and S. Padhy, A Course of Ordinary and Partial Differential Equations, Kalyani Publishers, New Delhi.

2. G.B. Thomas, M.D. Weir, J.R. Hass, Thomas’ Calculus, Pearson Publication.

3. R.G. Bartle, D.R. Sherbert, Introduction to Real Analysis, Wiley Publication

Applications of First order linear differential equations( Kirchhoff's law)

Kirchhoff's law

Second order linear homogeneous differential equations

Second order linear homogeneous differential equations

Solution of Second order linear homogeneous differential equations

Second order linear homogeneous differential equations

Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations

Second order linear non-homogeneous differential equations

Second order linear non-homogeneous differential equations

Finding particular integral consisting of exponential

Finding particular integral consisting of exponential

Finding particular integral consisting of trigonometric (Sine, cosine) using inverse operator method

https://www.youtube.com/watch?v=r6mkGRLpm_w

**Project-3 (2 hrs)**

Simple mass-spring system

https://www.youtube.com/watch?v=FJBPNJR2QJU

Linear system of equations by Gauss elimination method

Linear algebra

https://www.youtube.com/watch?v=2j5Ic2V7wq4

Linearly independent and dependent of a matrices, Rank of a matrix

http://courseware.cutm.ac.in/wp-content/uploads/2020/06/Linear-Algebra-3.pdf

https://www.youtube.com/watch?v=uQhTuRlWMxw

https://www.youtube.com/watch?v=k7RM-ot2NWY

**Project-4 (Flip Class) (2 hrs)**

Report on finding the traffic flow in the net of one-way streets

Determinants of a matrix, Cramer’s Rule

Determinant

Linear algebra

https://www.youtube.com/watch?v=Ip3X9LOh2dk

Conditions of Existence and Uniqueness of Solutions

https://www.slideshare.net/Pokar/system-of-linear-equations-45700370

https://www.youtube.com/watch?v=8JhhMhN_45Q

Properties of Eigenvalues and Eigen Vectors

Eigen values and eigen functions

https://www.youtube.com/watch?v=8F0gdO643Tc

Some application problems of Eigenvalues and Eigen Vectors

Eigen values and eigen functions

https://www.youtube.com/watch?v=R13Cwgmpuxc

**Project-5 (Flip Class) (2hrs)**

(i)Find the limit states of the Markov process model.

(ii)Find the growth rate in the Leslie model

Symmetric, Skew-Symmetric and Orthogonal Matrices

Symmetric matrix

https://www.youtube.com/watch?v=v5abfTlztTc

Properties of Symmetric, Skew-Symmetric and Orthogonal Matrices

https://www.youtube.com/watch?v=v5abfTlztTc

Problems of Symmetric, Skew-Symmetric and Orthogonal Matrices

https://www.youtube.com/watch?v=v5abfTlztTc

**Project-6 (Flip class) (2 hrs)**

To make a report to show that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations?

Mrs. Saubhagyalaxmi Singh works as an Assistant Professor, Department of Mathematics, Centurion University of Technology and Management. She has over 6 years of teaching experience with exposure to Utkal University, Bhubaneswar, Department of Mathematics. She has published 6 papers communicated to different International Journals. She has guided 7 M.Sc. projects in her area. She has […]

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