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An Introduction to Differential Equations by Ghosh and Maity
Differential equations are mathematical equations that relate one or more unknown functions and their derivatives. They are widely used in various fields of science and engineering to model natural phenomena, such as heat transfer, fluid dynamics, population growth, and so on. A solution to a differential equation is a function that satisfies the equation when substituted into it.
An Introduction to Differential Equations by Ghosh and Maity is a textbook that covers the basic concepts and methods of solving ordinary differential equations of first and second order. It also introduces some topics in partial differential equations, such as the heat equation, the wave equation, and Laplace's equation. The book is intended for undergraduate students of mathematics, physics, and engineering. It provides numerous examples and exercises to illustrate the theory and applications of differential equations.
The book is divided into 12 chapters. The first chapter gives an overview of differential equations and their classification. The second chapter deals with the method of separation of variables for solving first order differential equations. The third chapter introduces some special types of first order differential equations, such as exact, homogeneous, linear, Bernoulli, and Riccati equations. The fourth chapter discusses the existence and uniqueness of solutions of first order differential equations using the Picard-LindelÃ¶f theorem and its applications.
The fifth chapter covers the theory and methods of solving second order linear differential equations with constant coefficients. It also explains how to find particular integrals by the method of undetermined coefficients and by the method of variation of parameters. The sixth chapter extends the results of the previous chapter to second order linear differential equations with variable coefficients. It also introduces some special functions that arise as solutions of these equations, such as Bessel functions, Legendre polynomials, and Hermite polynomials.
The seventh chapter studies some applications of second order linear differential equations in physics and engineering, such as simple harmonic motion, damped oscillations, forced vibrations, electric circuits, and mechanical systems. The eighth chapter introduces some basic concepts and techniques of Laplace transform and its use in solving linear differential equations with constant coefficients. The ninth chapter presents some topics in systems of linear differential equations with constant coefficients, such as matrix methods, eigenvalue problems, phase portraits, and stability analysis.
The tenth chapter deals with some topics in nonlinear differential equations, such as autonomous systems, equilibrium points, linearization, Lyapunov stability, limit cycles, and bifurcations. The eleventh chapter introduces some concepts and methods of partial differential equations, such as boundary value problems, separation of variables, Fourier series, Fourier transform, Sturm-Liouville theory, and Green's functions. The twelfth chapter discusses some applications of partial differential equations in physics and engineering, such as heat conduction, wave propagation, potential theory, and diffusion processes.
The book is well-written and comprehensive. It provides clear explanations and proofs of the main results. It also contains many worked-out examples and exercises that help the reader to understand and apply the concepts and methods of differential equations. The book is suitable for a one- or two-semester course on differential equations for undergraduate students. 061ffe29dd