Higher Engineering Mathematics oeNUMERICAL METHODS IN ENGINEERING AND SCIENCE for BE., B. Tech., ME., M.C.A. B. (Computer Science), M.Sc. (Physics /Maths) — (10th Edition) i See * Numerical Solution of Algebraic, Transcendental and Simultaneous Equations * Matrix Inversion and Eigen-value Problems + Empirical Laws and Curve Fitting + Finite Differences and Interpolation + Numerical Differentiation and Integration * Difference Equations * Numerical Solution of Ordinary and Partial Differential Equations © Linear Programming * Use of Computers in Numerical Methods + Numerical Techniques using C, C++ & MATLAB ELEMENTARY ENGINEERING MATHEMATICS for I & II Semesters of B.Tech. & Diploma Courses—(15th Edition) * Preliminaries, Solution of Equations & Curve Fitting + Determinants, Matrices & Complex Numbers + Analytical Plane & Solid Geometry * Differential and Integral Calculus with Applications * Partial Differentiation & its Applications * Multiple Integrals and Beta, Gamma Functions * Vector Analysis * Ordinary Differential Equations with Applications + Partial Differential Equations with Applications + Infinite Series and Fourier Series + Laplace Transforms + Simple Numerical Methods + Elements of Statistics & Probability * Introduction to Linear ProgrammingProfessor of Applied Mathematics Principal Scientifie/ Officer (Ex,) Defence Research & Debelopmént Organisation, New Delhi Formerly of Colla of Military Engineering, Poona, Delhi College of Engineering, Delhi M.LE,, I. Engr. (U.K), MJ. Mar. EST. Associate Professor LG. Delhi Technical University for Women, Delhi ® Investing in Learing® Operational Office , Onkar House, Ground Floor, Opp. Happy School, Daryaganj, New Delhi-11002 011-45033819 » Mob. 09811541460 contactus@khannapublishers.inPublished by : Romesh Chander Khanna & Vineet Khanna for KHANNA PUBLISHERS 2B, Nath Market, Nai Sarals, ‘Delhi-110006 (India). Visit us at : www-khannapublishers.in Copyright : Author and Publishers Jointly ©1979 and onward. This book or part there of cannot be translated or reproduced in any form without the written permission of both Authors and the Publishers. The right to translation, however, reserved with the Authors alone, Hologram & Description ‘To all readers of our books, to prevent yourself from being defrauded by piraters. Please make sure that there is an Hologram on the cover of our books with the below specifications. If you find any book without the Hologram or Description, please mail us at contactus@khannapublishers.in. Thankingyou 2 Channel Effect xP Kinetic Effect: 0123457 Laser Numbering Fu Olso0 '\ Khanna’ Emboss Te ‘ Publishere*—— Gradient effect Guilloche Etfect ge >= GENUIN, Modulated Depth ISBN No. : 978-81-938284.9-1 First Bdition : 1965, August Forty Fifth Bdition : 2024 Price: 1179.00To who served the cause of education with a missionary zeal for over five decades> Preface to the 45th Edition ‘The book with a legacy, has run through its 44th edition, with twenty eight subsequent reprints, which itselfis a testament toits enduring popularity. Despite maintaining the familiar format and features that have garnered such widespread acclaim, we have meticulously revised the text, introducing additions and alterations to enhance its utility. We have expanded the number of solved examples, integrating problems sourced from the latest university question papers. It is our sincere hope that these revisions will further elevate the book's standing amongitsreaders. ‘The authors extend their heartfelt thanks to the esteemed readers for their letters of appreciation and to the renowned professors for their invaluable suggestions and unwavering support. In particular, they express gratitude to Dr. B.B. Singh of Vidyavihar, Ionere (Raigarh); Dr. V. Vasudeva Murthy of S.R.K.R. Engineering College, Bhimavaram (A.P.); Prof. P. Veeraiah of Sri Venkateswara College of Engineering, Sriperumbudur (T.N.); Dr. Jeavargi Phakirappa of R.Y.M. Engineering College, Bellary (Ka.); Dr. T. Subedi of the National Institute of Technology, Shillong; Dr. 8. Mayilaganan of S.R.M. University, Kuttankulathur (T.N.); Dr. Garima Aggarwal of Manipal University, Jaipur; Dr. Barun Kumar Yadav, Head of the Department of Mathematics at Chouksey Engineering College, Bilaspur (C.G.); Dr. Arvind Kumar Sinha of the National Institute of Technology, Raipur; Dr. Rashmi Bhardwaj of G.G.S. Indraprastha University, Delhi; Dr. T. Yogalakshmi of Vellore Institute of Technology, Vellore; Dr. C.N.B. Rao of S.R.KR. Engineering College, Bhimavaram (A.P.); Dr. V. Dhanapalan of Government College of Technology, Pudar (T-N.); and Prof. Shivani Mehra of Chandigarh University, Mohali. For the present 45% edition of the book, the authors express their sincere gratitude to Dr. K. L. Sai Prasad, Associate Professor of Mathematics and Head of the Department of Basic Sciences and Humanities at G.V.P. College of Engineering for Women, Visakhapatnam. His insightful contributions and addition of relevant questions and materials during the editing, have been incorporated into this current edition. Suggestions for improvement of the text and intimation of misprints will be thankfully acknowledged. New Delhi B.S. GREWAL JS. GREWAL ‘LK. DHANOAContents sr TSS PN L a a eres 9. 10. Solution of Equations Linear Algebra : Determinants, Matrices Vector Algebra & Solid Geometry CE Differential Calculus and Its Applications Partial Differentiation and Its Applications Integral Caleulus and Its Curve Tracing ‘Multiple Integrals and Beta, Gamma Functions ‘Vector Caleulus and Its Applications Infinite Series Fourier Series and Harmonic Analysis 116 17-88 39-148 149-216 217-271 272314 315364 365422 423464 465509 Daas u. 2 13, “a 15, Differential Equations of First Order Applications of Differential Equations of First Order Linear Differential Equations Applications of Linear Differential Equations Differential Equations of Other Types Series Solution of Differential Equations and Special Functions Partial Differential Equations Applications of Partial Differential Equations (vi) 510541 542563 564-596 597625 626-638 639-679 680—706 107747Tae 19. Complex Numbers and Functions 20. Calculus of Complex Functions TENS 21, Laplace Transforms - 839882 Fourier Transforms 883912 23, Z:Transforms 913-931 Rarer =e 24, Empirical Laws and Curve-fitting, (932—949 Serene Ba escB227 26, Probability and Distributions: oe 981—1022 Sy emerge eet 28, Numerical Solution of Equations: (1045—1076 £2 eLearn eon SEED ee er » OERAT cer ere a er ee ieee e oe Se ee ee Sra ee ae Sr 3A, Linear Programming —— 35. Catelu of Veriaiona red re ea 37. Discrete Mathematics ws 1292—1323 eA eens raz Appendix 1: UnstalInormatien © 15451969 Geena renia ia Taw Coote - am Table I: Bes Functions 7 1371 Toil Laplace Tranatrms a ee 7 at ees = Table VI; Values of x? .” 1376 Table VII; Values of F . ‘1377, Appendix 3: Answers to Problems we 1378-1457 — ees i)Note : The references given alongside the problems pertain to the Degree Engineering Examinations of the various universities and professional bodies. The abbreviations used for same ofthese are given below : Ando stands for Andhra. University, Waltir Anna . ‘Anna University, Chennai ‘Bhopal . Rajiv Gandhi Technical University, Bhopal aru. . ‘ju Patnaik Technical University, Rourkela csunu. . (Chhattisgarh Swami Vivekanand Technical University, Raipur Coimbatore . Bharathiyar University, Coimbatore beth . Guru Gobind Singh Indreprastha University, eli oru. . Debi Technological Univesity (Formerly Delhi Collage of Engineering) Hozaribag . Vinoba Bhave University, Hazaribag NTU, . slawahar Lal Nehmu Tecnological Univesity, Hyarabad Kru, . [APS Abdul Kalam Kerala Technological University, Thiuananthapuram urushetra . National Institute of Technology, Kurukshetra Marathwade . BAM. University, Aurangebad Pru, . Punjab Technical University, Jalandhar ATU. . Rajasthan Technical University, Kota oktok . Maharshi Dayanané University, Rohtak ueru, . UP Technical University, Lucknow uKru, . Uttarakhand Technical University, Dehradun wru. . Visveswaraiah Technological University, Belgaum Worangal . Warangal University of Technology, Warangal ws.ru, . Maulana Abdul Kalam Azad, West Bengal Unversty of Technology, Kolkata og)