Differential Equations:

Cover......Page 1 Contents......Page 4 Preface......Page 12 About the Author......Page 13 1.1.1 Differential Equation......Page 14 1.2.1 Formation of a Differential Equation......Page 17 1.2.2 Solution of a Differential Equation......Page 25 Exercise 1.1......Page 26 2.1 First order and first degree differential equations......Page 30 2.1.1 Variable Separable Equation......Page 31 Exercise 2.1......Page 36 2.1.2 Homogeneous Equations......Page 38 Exercise 2.2......Page 43 2.1.3 Non-homogeneous Equations......Page 45 Exercise 2.3......Page 59 2.1.4 Exact Equations......Page 61 Exercise 2.4......Page 68 2.1.5 Inexact Equation—Reducible to Exact Equation by Integrating Factors......Page 69 Exercise 2.5......Page 73 Exercise 2.6......Page 83 2.1.6 Linear Equations......Page 85 Exercise 2.7......Page 95 2.1.7 Bernoulli’s Equation......Page 97 Exercise 2.8......Page 101 2.2 Applications of ordinary differential equations......Page 103 Exercise 2.9......Page 107 2.2.1 Geometrical Applications......Page 108 Exercise 2.10......Page 115 3.1 Introduction......Page 116 3.1.1 Linear Differential Equations of the Second Order......Page 117 3.1.2 Homogeneous Equations—Superposition or Linearity Principle......Page 119 3.1.3 Fundamental Theorem for the Homogeneous Equation......Page 120 3.1.4 Initial Value Problem (IVP)......Page 121 3.1.5 Linear Dependence and Linear Independence of Solutions......Page 123 3.1.7 Second Order Linear Homogeneous Equations with Constant Coefficients......Page 124 Exercise 3.1......Page 128 3.1.8 Higher Order Linear Equations......Page 129 3.1.9 Linearly Independent (L.I.) Solutions......Page 130 3.1.10 Exponential Shift......Page 132 Exercise 3.2......Page 137 3.1.11 Inverse operator D−1 or 1D......Page 139 3.1.12 General Method for Finding the P. I.......Page 140 Exercise 3.3......Page 142 3.2 General solution of linear equation f (D) y = Q(x) 3-29......Page 144 Exercise 2.4......Page 150 3.2.1 Short Methods for Finding the Particular Integrals in Special Cases......Page 151 Exercise 3.5......Page 157 Exercise 3.6......Page 165 Exercise 3.7......Page 168 Exercise 3.8......Page 171 Exercise 3.9......Page 176 3.2.2 Linear Equations with Variable Coefficients— Euler–Cauchy Equations (Equidimensional Equations)......Page 178 Exercise 3.10......Page 184 3.2.3 Legendre’s Linear Equation......Page 186 Exercise 3.11......Page 188 3.2.4 Method of Variation of Parameters......Page 189 Exercise 3.12......Page 196 3.2.5 Systems of Simultaneous Linear Differential Equations with Constant Coefficients......Page 197 Exercise 3.13......Page 201 4.1 Equations solvable for p......Page 204 Exercise 4.1......Page 213 4.2 Equations solvable for y......Page 214 Exercise 4.2......Page 222 4.3 Equations solvable for x......Page 223 Exercise 4.3......Page 229 Chapter 5: Linear Equation of the Second Order with Variable Coefficients......Page 230 5.1 To find the integral in C.F. by inspection, i.e. to find a solution of......Page 232 Exercise 5.1......Page 245 5.2 General solution of R by changing the dependent variable and removing the first derivative (Reduction to normal form)......Page 247 Exercise 5.2......Page 252 5.3 General solution of by changing......Page 254 Exercise 5.3......Page 261 6.1.2 Power Series Method of Solution of Linear Differential Equations......Page 264 6.1.3 Existence of Series Solutions: Method of Frobenius......Page 265 6.1.4 Legendre Functions......Page 267 6.1.5 Legendre Polynomials Pa(x)......Page 269 6.1.6 Generating Function for Legendre Polynomials Pn(x)......Page 275 6.1.7 Recurrence Relations of Legendre Functions......Page 276 6.1.9 Orthogonality of Legendre Polynomials Pn(x)......Page 280 6.1.11 Christoffel’s Expansion......Page 283 6.1.12 Christoffel’s Summation Formula......Page 284 6.1.13 Laplace’s First Integral for (x)......Page 285 6.1.14 Laplace’s Second Integral for (x)......Page 286 6.1.15 Expansion of f(x) in a Series of Legendre Polynomials......Page 287 Exercise 6.1......Page 291 6.2.2 Bessel Functions......Page 292 6.2.3 Bessel Functions of Non-integral Order p: Jp(x) and J-p(x)......Page 294 6.2.4 Bessel Functions of Order Zero and One: J0(x), J1(x)......Page 295 6.2.5 Bessel Function of Second Kind of Order Zero Y0(x)......Page 296 6.2.6 Bessel Functions of Integral Order: Linear Dependence of Jn(x) and J-n(x)......Page 297 6.2.8 Generating Functions for Bessel Functions......Page 298 6.2.9 Recurrence Relations of Bessel Functions......Page 300 6.2.10 Bessel’s Functions of Half-integral Order......Page 302 6.2.11 Differential Equation Reducible to Bessel’s Equation......Page 304 6.2.12 Orthogonality......Page 305 6.2.13 Integrals of Bessel Functions......Page 308 6.2.14 Expansion of Sine and Cosine in Terms of Bessel Functions......Page 309 Exercise 6.2......Page 315 6.3 Chebyshev polynomials......Page 316 Exercise 6.3......Page 329 7.2.1 Laplace Transform......Page 330 7.3 Fourier integral theorem......Page 331 7.3.1 Fourier Sine and Cosine Integrals (FSI’s and FCI’s)......Page 332 7.4.1 Fourier Integral Representation of a Function......Page 333 7.5 Fourier transform of f (x)......Page 334 7.5.1 Fourier Sine Transform (FST) and Fourier Cosine Transform (FCT)......Page 335 7.6 Finite Fourier sine transform and finite Fourier cosine transform (FFCT)......Page 336 7.6.1 FT, FST and FCT Alternative definitions......Page 337 7.7.2 Convolution Theorem......Page 338 7.8.1 Linearity Property......Page 339 7.8.3. Shifting Property......Page 340 7.8.4 Modulation Theorem......Page 341 Exercise 7.1......Page 358 7.9 Parseval’s identity for Fourier transforms......Page 360 7.10 Parseval’s identities for Fourier sine and cosine transforms......Page 361 Exercise 7.2......Page 364 8.2.1 Order......Page 366 8.2.3 Homogeneity......Page 367 8.3 Origin of partial differential equation......Page 368 8.4 Formation of partial differential equation by elimination of two arbitrary constants......Page 369 Exercise 8.1......Page 373 8.5 Formation of partial differential equations by elimination of arbitrary functions......Page 374 Exercise 8.2......Page 379 8.6.1 Linear Equation......Page 380 8.7 Classification of solutions of first-order partial differential equation......Page 381 8.7.2 General Integral......Page 382 8.8 Equations solvable by direct integration......Page 383 Exercise 8.3......Page 386 8.9 Quasi-linear equations of first order......Page 387 8.10.2 Two Variables are Separable......Page 389 8.10.3 Method of Multipliers......Page 391 Exercise 8.4......Page 401 8.11 Non-linear equations of first order......Page 403 Exercise 8.5......Page 412 8.12 Euler’s method of separation of variables......Page 413 Exercise 8.6......Page 418 8.13.1 Introduction......Page 419 8.13.2 Classification of Equations......Page 420 8.13.4 Solution of One-dimensional Heat Equation (or diffusion equation)......Page 421 Exercise 8.7......Page 435 8. 13.6 Vibrating String with Zero Initial Velocity......Page 437 8.13.7 Vibrating String with Given Initial Velocity and Zero Initial Displacement......Page 443 8.13.8 Vibrating String with Initial Displacement and Initial Velocity......Page 450 Exercise 8.8......Page 452 8.13.9 Laplace’s equation or potential equation or two-dimensional steady-state heat flow equation equation......Page 453 Exercise 8.9......Page 461 Exercise 8.10......Page 469 Index......Page 470 Differential Equations presents the basics of differential equations, adhering to the UGC curriculum for undergraduate courses on differential equations offered by all Indian universities. With equal emphasis on theoretical and practical concepts, the book provides a balanced coverage of all topics essential to master the subject at the undergraduate level, making it an ideal classroom text. Written in lucid, easy-to-understand language, the topics discussed in this student-friendly book are amply supported by exhaustive number of problems as well as over 300 solved examples and 400 end-of-chapter exercises This book presents the basics of differential equations, adhering to the UGC curriculum for undergraduate courses on differential equations offered by all Indian universities. With equal emphasis on theoretical and practical concepts, the book provides a balanced coverage of all topics essential to master the subject at the undergraduate level, making it an ideal classroom text. Written in lucid, easy-to-understand language, the topics discussed in this student-friendly book are amply supported by exhaustive number of problems as well as over 300 solved examples and 400 end-of-chapter exercises Formation of a differential equation Differential equations of first order and first degree Linear differential equations with constant coefficients Differential equations of the first order but not of the first degree Linear equation of the second order with vairable coeffecients Integration in series : Legendre, Bessel, and Chebyshev functions Fourier integral transforms Partial differential equations.