'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. |
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 Book:
- Advanced Engineering Mathematics, Erwin Kreyszig, 10th Edition, Wiley, 2016.
Reference Book:
- Engineering Mathematics, Srimanta Pal and Subodh C Bhunia, Oxford university press, 2015.
- Advanced Engineering Mathematics, Wylie and Barrett, 6th Edition, McGraw Hall India, 2015.
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