COURSE SUMMARY
Course Title:
Differential Equations and Transforms
Course Code:
19MAT114
Year Taught:
2019
Semester:
2
Degree:
School:
School of Engineering
Campus:
Bengaluru
Chennai
Coimbatore
Amritapuri

'Differential Equations and Transforms' is a course offered in the second semester of  B. Tech. programs at the School of Engineering, Amrita Vishwa Vidyapeetham.

### Course Outcomes

At the end of the course the student will be able to

 CO1 To frame and solve homogeneous and non-homogeneous ordinary differential equations corresponding to different p ractical scenarios. CO2 Apply the geometric state space approach to the analysis of ODEs to understand qualitative features. CO3 To find the Fourier series of arbitrary functions and to find the Fourier and Laplace transforms of functions. CO4 Recognize the three basic types of partial differential equations and to use both analytic methods to the solution of hyperbolic, parabolic and elliptic partial differential equations.

#### SYLLABUS

First order ODE : Ordinary Differential Equations – Basic concepts, modelling, first order ODEs, exact ODEs, integrating factors. ( 5 hrs )

Second order ODE : , homogeneous linear ODEs, Euler-Cauchy equations, existence and uniqueness of solution, Wronskian, non-homogeneous ODEs, variation of parameters. Modelling of free and forced oscillations of spring-mass system, resonance. ( 1 3 hrs )

Higher order ODEs, homogeneous and nonhomogeneous linear ODEs. System of ODEs – Phase space, velocity field, flow, fixed points, stability of fixed points. Qualitative methods for ODEs.  ( 1 2 hrs )

Fourier Series, arbitrary period, even and odd expressions, half range expressions, Fourier Integral, Fourier transforms. Laplace transform, transform of derivatives and integrals, solution of initial value problems by Laplace transform. ( 1 5 hrs )

Partial differential equations – Basics of PDEs. Modelling of vibrating string, wave equation, solution by separation of variables, D’Alembert’s solution, Heat flow modelling, heat equation, solution of heat equation by Fourier series, heat equation in very long bars, solution by Fourier transforms, Laplace’s equation and its solution.  ( 1 5 hrs )

Course Evaluation Pattern:

Test-1 -15 marks (two hour test)

CA - 20 marks (Quizzes / assignments / lab practice) Test – 2- 15 marks (two-hour test)

End semester- 50 marks. Total - 100 marks.

Supplementary exam for this course will be conducted as a three-hour test for 50 marks.

#### TEXT BOOKS / REFERENCES

Text Book:

1. Advanced Engineering Mathematics, Erwin Kreyszig, 10th Edition, Wiley, 2016.

Reference Book:

1. Engineering Mathematics, Srimanta Pal and Subodh C Bhunia, Oxford university press, 2015.
2. Advanced Engineering Mathematics, Wylie and Barrett, 6th Edition, McGraw Hall India, 2015.