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Hello! Let’s have a look at an area of Mathematics that can be rightly described as the language of the Sciences…
Differential Equations can describe how populations change, how heat moves, how springs vibrate, how radioactive material decays and much more. They are a very natural way to describe many things in the universe.
Course Code: MM5CRT06
Course Credit: 4
Course Type: Core
A differential equation is an equation that relates one or more functions and their derivatives. In applications in the real world, differential equations define a relationship between physical quantities.
Such relations are common, therefore differential equations play a prominent role in many disciplines including engineering, physics, economics and biology.
Hence, Differential Equations can be regarded as an essential course not only for a degree in Mathematics, but also for any field that studies how things work!
- To identify and solve first order differential equations including separable, homogeneous, exact, and linear.
- To solve second order and higher order linear differential equations.
- To solve differential equations using variation of parameters.
- To find series solutions of first and second order differential equations.
- To solve first order differential equations in 3 variables ( of the form )
- To understand the origin of first order linear partial differential equations and solve some special types using Lagrange’s method.
Learning Outcomes :
After the completion of the course, students will be able to
1.Explain the concept of differential equation, classify them and to solve first order differential equations utilizing the standard techniques for separable, exact, linear, homogeneous, or Bernoulli cases.
- Find the complete solution of a non homogeneous differential equation as a linear combination of the complementary function and a particular solution.
- Find the complete solution of a non homogeneous differential equation with constant coefficients by the method of undetermined coefficients and by variation of parameters.
- Recognize and classify partial differential equations and to find integral surfaces passing through a given curve, for certain types.
- G.F. Simmons, S.G. Krantz – Differential Equations, (Tata McGraw Hill-New Delhi). (Walter Rudin Student Series)
- Ian Sneddon – Elements of Partial Differential Equation (Tata Mc Graw Hill)
- Shepley L Ross – Differential Equations, ( Wiley Student Edition)
- George F Simmons – Differential Equations with Applications and Historical Notes ( Tata McGraw Hill)
- Erwin Kreyszig-Advanced Engineering Mathematics, 10th Edition, Wiley
Studies on differential equations consist of two parts – one part consists of the study of their solutions (the set of functions that satisfy each equation), and another part of the properties of their solutions. We concentrate on the former here.The course commences with a recap of the basics .It moves on to the solution of various special kinds of first and higher order differential equations, and finally the concept of partial differential equations and solutions of Lagrange’s equation. Besides regular lectures, there will be sessions for problem solving, seminars and discussions too.
The internal evaluation is based on the participant’s participation, assignments, seminars, attendance, internal assessments and an external end semester exam.
The external end semester exam carries 80 marks whereas the internal component carries 20 marks.
Topics for the assignments will be either announced in the online/offline class during course or put in the virtual classroom.
What is a differential equation ?: The nature of solutions, Separable equations, First order linear equations, Exact equations, Orthogonal trajectories and families of curves, Homogeneous equations, Integrating factors, Reduction of order-dependent variable missing-independent variable missing Text 1: Chapter 1 (Sections 1.2 to 1.9)
Second order linear equations: Second order linear equations with constant coefficients (which includes Euler’s equidimensional equations given as exercise 5 in page 63 of Text 1), The method of undetermined coefficients, The method of variation of parameters, The use of a known solution to find another, Higher order linear equations Text 1: Chapter 2 ( Sections 2.1, 2.2, 2.3, 2.4, 2.7) (example 2.17 is excluded ).
Power Series solutions and special functions: Series solutions of first order differential equations, Second order linear equations: ordinary points (specially note Legendre’s equations given as example 4.7), Regular singular points, More on regular singular points. Text 1: Chapter 4 ( Sections 4.1, 4.2, 4.3, 4.4 and 4.5 ).
Partial Differential equations : Methods of solution of dx/P = dy/Q = dz/R , Origin of first order partial differential equations, Linear equations of the first order, Lagrange’s Method (proof of theorem 2 and theorem 3 are excluded) Integral surfaces passing through a given curve. Text 2: Chapter 1 ( Section 3) and Chapter 2 ( Sections 1, 2 ,4 (no proof of theorem 2 and theorem 3) and 5)
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