Change is the law of nature. We all know that most things evolve with time. They are also diverse and non-uniform in space. The evolution in space and time is best described in a quantitative way in terms of differential equations. The oscillations of a simple pendulum, as in an old fashioned wall clock, propagation of sound and light waves enabling us to hear and see, excitation and deexcitation of electrons in atoms which generate light waves, complicated interactions in chemical reactions, growth of bacteria feeding on rotten foodstuff, foxes eating rabbits who eat grass, and similar other physical problems can be modelled by differential equations.
In fact, most of the natural laws in physics, chemistry, life sciences, astronomy, engineering etc. find their most natural expression in the language of differential equations. It is for these reasons that in this 2-credit course IGNOU have focused our attention entirely on differential equations. The course is presented in two blocks. The first block deals with ordinary differential equations, i.e., differential equations in which the unknown function depends only on one variable. Differential equations involving unknown functions of more than one variable are called partial differential equations. These form the subject of Block 2. Our emphasis in this course is on studying the methods of solving ordinary and partial differential equations, with particular reference to their applications in physics.
Syllabus Per Block Divisions in IGNOU BSc Physics - Mathematical Methods in Physics-II PHE-05
Block 1: Ordinary Differential Equations
- Unit 1: First Order Ordinary Differential Equations
- Unit 2: Second Order Ordinary Differential Equations with Constant Coefficients
- Unit 3: Second Order Ordinary Differential Equations with Variable Coefficients
- Unit 4: Some Applications of ODEs in Physics
Detailed Syllabus for IGNOU BSc Physics - Ordinary Differential Equations
Classification, General and Particular Solution, Existence and Uniqueness of a Particular Solution, General Properties of the Solutions of Linear ODEs; First Order Ordinary Differential Equations: Equations Reducible to Separable Form: Method of Separation of Variables, Homogeneous First Order ODEs; Exact Equations, First Order Linear Differential Equations; Equations Reducible to First Order; Second Order Ordinary Differential Equations with Constant and Variable Coefficients: Basic Terminology; Homogenous and Inhomogenous Linear Equations with Constant Coefficients, The Method of Undetermined Multipliers; Method of Variation of Parameters, Complementary Functions and Particular Integral, Linear Independence, Wronskian, Power Series Solution, Frobenius Method; Applications of ODEs in Physics: Mathematical Modelling; First Order ODEs in Physics: Applications in Newtonian Mechanics, Simple Electrical Circuits; Second Order ODEs in Physics: Rotational Mechanical Systems, Planetary Orbits; Coupled Differential Equations: Coupled Oscillators, Coupled Electrical Circuits, Charged Particle Motion in Electric and Magnetic Fields.
Block 2: Partial Differential Equations
- Unit 5: An Introduction to Partial Differential Equations
- Unit 6: Partial Differential Equations in Physics
- Unit 7: Fourier Series
- Unit 8: Applications of Fourier Series to PDEs
Detailed Syllabus for IGNOU BSc Physics - Partial Differential Equations
Functions of More than One Variable; Limits and Continuity, Partial Differentiation, Differentiability; Classification of Partial Differential Equations, General and Particular Solution; Partial Differential Equations in Physics: Method of Separation of Variables, Solution of Initial and Boundary Value Problems; Fourier Series: The Need for Fourier Series, Determination of Fourier Coefficients, Use of Fourier Series: as an Approximation, Even and Odd Functions, Fourier Sine and Cosine Series, Half- range Expansions, The Convergence of Fourier Series; Applications of Fourier Series to Partial Differential Equations: Diffusion Equation: Heat conduction and Diffusion of Particles; Wave Equation, The Plucked String Problem, Torsional Vibrations; Laplace's Equation, Steady State Heat Flow, the Potential Problem.