Engineering Mathematics with MATLAB® (Chul Ki Song Jong-Ryeol Kim) (z-library.sk, 1lib.sk, z-lib.sk)

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Engineering Mathematics with MATLAB® This textbook takes a streamlined, practical approach, designed to make engineering mathematics accessible and manageable for undergraduate stu- dents and instructors alike. Students will gain a fundamental understanding within the scope of a two-semester course. This textbook introduces students to the fundamental principles of en- gineering mathematics through concise explanations, systematically guiding them from the basics of frst-order, second-order, and higher-order ordinary diferential equations (ODEs), Laplace transforms, and series solutions of ODEs. It then transitions to more advanced topics, including linear algebra, linear system of ODEs, vector diferential calculus and vector integral cal- culus, Fourier analysis, partial diferential equations (PDEs), and concludes with complex numbers, complex functions, and complex integration. The book presents fundamental principles systematically with concise explana- tions. It features categorized key concepts, detailed solutions, and alternative methods to connect material to prior knowledge. Exercises are thoughtfully organized, balancing problem-solving practice with real-world applications in felds like mechanical engineering, electrical engineering, chemical engi- neering, and so on. Notably, this book incorporates MATLAB® to enhance understanding. MATLAB®-based examples simplify complex calculations, ofering visualizations that connect theory and practice. Chapters also include optional advanced topics, providing deeper insights for motivated learners. Designed with practicality in mind, this book ofers a balanced approach to mastering engineering mathematics, with a manageable workload aligned to academic schedules. It is an invaluable resource for instructors seeking ef- fective teaching tools and for students aiming to build strong mathematical foundations that they can apply to their own engineering discipline. Chul Ki Song is an Emeritus Professor at the School of Mechanical Engineer- ing, Gyeongsang National University, Korea. He earned his Ph.D. degree from Seoul National University, Korea. He previously worked for KIA Motors. Jong-Ryeol Kim is a Professor in the Department of Electrical Engineering, Sejong University, Korea. He earned his Ph.D. degree from KAIST, Korea. He previously worked for Samsung Electronics.

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Engineering Mathematics with MATLAB® Chul Ki Song and Jong-Ryeol Kim

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Designed cover image: Engineering 3d Computer Aided Design Concept With Metallic Gears On A Virtual Reality Blueprint Surface With Glowing Programming Code And Infographic Overlay High-Res Stock Photo - Getty Images MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® or Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software. First edition published 2026 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2026 Chul Ki Song and Jong-Ryeol Kim Reasonable eforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microflming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978- 750-8400. For works that are not available on CCC please contact mpkbookspermissions@tandf.co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identifcation and explanation without intent to infringe. ISBN: 9781032979847 (hbk) ISBN: 9781041002550 (pbk) ISBN: 9781003608912 (ebk) DOI: 10.1201/9781003608912 Typeset in Sabon by KnowledgeWorks Global Ltd.

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Contents Preface xiii 1 First-order ordinary diferential equations 1 1.1 Diferential equations and their solutions 1 1.1.1 Fundamentals of diferential equations 1 1.1.2 First-order diferential equations 1 1.1.3 Ordinary diferential equations and partial diferential equations 1 1.1.4 Representation of diferential equations 2 1.1.5 Diferential equations and solutions 2 1.1.6 Initial value problems 4 1.1.7 Diferentiation 5 1.1.8 Integration 8 1.2 Separable frst-order ODEs 13 1.2.1 Method of separating variables 13 1.2.2 Extended method of separating variables 16 1.3 Exact frst-order ODEs and integrating factors 19 1.3.1 Exact ODE 19 1.3.2 Exact ODE using integrating factors 25 1.3.3 How to fnd integrating factors 27 1.4 General solution of the frst-order ODE 33 1.4.1 Standard form of a frst-order ODE 33 1.4.2* Bernoulli equation (*optional) 36 1.5 Application of frst-order ODEs 40 1.5.1 Heat transfer (Newton’s law of cooling) 40 1.5.2 Decay of a radioactive element 42 1.5.3 Electric circuit 43

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vi Contents 1.6* Numerical methods for ODEs (*optional) 51 1.6.1 Euler method 51 1.6.2 Runge-Kutta method 53 1.7 Utilizing MATLAB® 54 1.7.1 Starting with MATLAB® 54 1.7.2 To create an m-fle in MATLAB® 56 1.7.3 Plotting the graph of functions 56 1.7.4 Plotting the solution graph of the ODEs by using Runge-Kutta method 57 1.7.5 Solve the frst-order ODE by using MATLAB® 60 2 Second-order ordinary diferential equations 71 2.1 Homogeneous ODEs 71 2.1.1 Standard form of the second-order ODEs 71 2.1.2 Linear combination, basis, and general solution 71 2.1.3 Initial value problem (IVP) 72 2.1.4 Find the second solution if one solution is known (method of variation of parameters) 72 2.2 Second-order homogeneous ODEs with constant coefcients 76 2.2.1 Characteristic equation 76 2.2.2 General solution 76 2.2.3 Diferential operators 79 2.3 Euler-Cauchy equations 80 2.3.1 Characteristic equation 80 2.3.2 General solution 81 2.4 Second-order nonhomogeneous ODEs 84 2.4.1 Method of undetermined coefcients 84 2.4.2 Method of variation of parameters 89 2.5 Application of second-order ODEs 93 2.5.1 Vibration 93 2.5.2 Electrical circuit 102 2.6 Utilizing MATLAB® 109 3 Higher-order ordinary diferential equations 126 3.1 Homogeneous ODE 126 3.1.1 Standard form of a homogeneous ODE 126 3.1.2 Bases and a general solution 127

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Contents vii 3.1.3 Third-order linear ODE 127 3.1.4 Fourth-order or higher-order linear ODE 128 3.1.5 Third-order or higher-order Euler-Cauchy equation 131 3.2 Nonhomogeneous ODE 135 3.2.1 Method of undetermined coefcients 135 3.3.2 Method of variation of parameters (Wronskian’s method) 139 3.3 Application of the higher-order ODEs 145 3.4 Utilizing MATLAB® 149 4 Laplace transforms 160 4.1 Defnition of the Laplace transform 160 4.1.1 Notation convention of Laplace transforms 160 4.1.2 Linearity of the Laplace transform 161 4.1.3 Laplace transform of 1 and tn 161 4.1.4 Laplace transforms of exponential function eat and trigonometric functions 162 4.1.5 s-shifting: Substituting s with s − a in the Laplace transform 164 4.1.6 Inverse Laplace transform 166 4.2 Laplace transforms of derivatives and integrals 167 4.2.1 Laplace transform of derivatives 167 4.2.2 Laplace transform of integral 170 4.2.3 Solution of the ODE with initial conditions 172 4.2.4 System analysis 176 4.2.5 (*optional) Solution of an ODE with t-shifted initial values 177 4.3 Unit step function and Dirac Delta function 181 4.3.1 Unit step function 181 4.3.2 t-shifting: Replacing t by t − a in f(t) 183 4.3.3 Dirac delta function 188 4.4 Convolution, integral equations, and diferentiation and integral of transforms 194 4.4.1 Convolution 194 4.4.2 Integral equations 197 4.4.3 Diferentiation of Laplace transforms 199 4.4 Integration of Laplace transforms 200 4.5 Application of Laplace transforms 203 4.6 Utilizing MATLAB® 211

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viii Contents 5 Series solutions of ordinary diferential equations 240 5.1 Power series method 240 5.1.1 Technique of the power series method 240 5.1.2 Theory of the power series method 243 5.2 Frobenius method 246 5.3 Bessel function and Legendre polynomials 257 5.3.1 Bessel function of the frst kind Jν (x) 257 5.3.2 Bessel function of the second kind Y0(x) 265 5.3.3 Bessel function of the second kind Y n(x) 269 5.3.4 Legendre polynomials 271 5.4 Utilizing MATLAB® 274 6 Linear algebra: Matrices, Determinants, and Eigenvalue Problems 284 6.1 Matrices fundamentals 284 6.1.1 Equality of matrices 285 6.1.2 Sum and diference of matrices, scalar multiplication 285 6.1.3 The product of two matrices 286 6.1.4 Transposition matrix 287 6.2 System of linear equations, determinants, and inverse matrix 289 6.2.1 System of linear equations 289 6.2.2 Linear system of equations: Gauss elimination 289 6.2.3 Linearly independent and rank of matrix 292 6.2.4 Determinant 294 6.2.5 Inverse matrix 295 6.2.6 Solution of a system of linear equations: Using inverse matrix 297 6.3 Eigenvalue problem (EVP) 300 6.3.1 Eigenvalues and eigenvectors for the basic form I in linear algebra 300 6.3.2 Eigenvalues and eigenvectors for the basic form II in linear algebra 302 6.3.3 Normalization 304 6.4 Applications of linear algebra 307 6.4.1 Statics 307 6.4.2 Vibration 308 6.4.3 Electrical circuit 310 6.5 Utilizing MATLAB® 312

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Contents ix 7 Linear system of ordinary diferential equations 324 7.1 System of two frst-order ODEs 324 7.1.1 Second-order ODE 324 7.1.2 Higher-order ODE 325 7.1.3 Homogeneous solution of the ODE 328 7.1.4 General solution of nonhomogeneous ODEs 334 7.2 Linear system of homogeneous ODEs 341 7.2.1 General solution for distinct real eigenvalues 342 7.2.2 General solution for double real eigenvalues 346 7.2.3 (*optional) General solution for complex conjugate eigenvalues 352 7.2.4 (*optional) Phase plane 355 7.3 Linear system of nonhomogeneous ODEs 360 7.4 Applications of a system of ODEs 367 7.4.1 Mixing in liquids 367 7.4.2 Electric circuit 371 7.5 Utilizing MATLAB® 377 8 Vector diferential calculus: Grad, Div, and Curl 382 8.1 Vectors operations 382 8.1.1 Unit vector 383 8.1.2 Vector representation using unit vectors 384 8.1.3 Magnitude of vector 384 8.1.4 Vectors operation 385 8.1.5 Inner product (dot product) 387 8.1.6 Application of the inner product 390 8.1.7 Outer product (cross product) 391 8.1.8 Application of the outer product 395 8.1.9 Scalar triple product 397 8.2 Derivatives of vector functions, direction of vector functions 399 8.2.1 Vector function 399 8.2.2 Chain rule 400 8.2.3 Derivative of a vector function 401 8.2.4 The vector equation of a curve with parametric representations 402 8.3 Gradient of a scalar feld, directional derivative 405 8.3.1 Gradient of a scalar feld 405 8.3.2 Surface normal vector 407 8.3.3 Directional derivative 409

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x Contents 8.4 Divergence and curl of a vector feld 412 8.4.1 Divergence of a vector feld 412 8.4.2 Curl of a vector feld 415 8.5 Utilizing MATLAB® 416 9 Vector integral and integral theorems 421 9.1 Line integrals 421 9.1.1 Defnition of line integrals 421 9.1.2 Path independence of line integrals 424 9.2 Double integrals 431 9.2.1 Variation of parameters 434 9.3 Green’s theorem in the plane 440 9.4 Surface integrals 446 9.4.1 Surface in surface integrals 446 9.4.2 Surface integrals 448 9.5 Triple integral and Gauss’s divergence theorem 453 9.6 Stokes’s theorem in the space 456 10 Fourier analysis 464 10.1 Fourier series 464 10.1.1 Function with period 2π 464 10.1.2 Function with period 2L 469 10.1.3 Even function and odd function 472 10.1.4 Half-range expansions 477 10.1.5 Orthogonality 481 10.1.6 (*optional) Orthogonality of eigenfunctions of Sturm-Liouville problems 483 10.2 Fourier integral 491 10.2.1 Defnition of Fourier integral 491 10.2.2 Fourier cosine integral and Fourier sine integral 494 11 Partial diferential equations 510 11.1 One-dimensional wave equation 510 11.1.1 Transverse vibration of string 510 11.1.2 Method of separation of variables 512 11.1.3 Solution of the transverse vibration of string 513 11.2 Two-dimensional wave equation 520 11.2.1 Wave equation of a rectangular membrane 520 11.2.2 Wave equation of a circular membrane 527

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Contents xi 11.3 Wave equation of sound pressure 533 11.3.1 (*optional) Sound pressure equation 533 11.3.2 Solution of the wave equation for sound pressure (plane wave) 535 11.3.3 Solution of sound pressure equation (spherical wave) 537 11.4 One-dimensional heat conduction equation 538 11.4.1 (*optional) Heat conduction equation 538 11.4.2 Method of separation of variables 540 11.4.3 Solution of the heat conduction equation 541 11.5 Two-dimensional heat conduction equation 545 11.5.1 (*optional) Heat conduction equation 545 11.5.2 Method of separation of variables 545 11.5.3 Solution of the two-dimensional heat conduction equation 546 12 Complex numbers and functions 552 12.1 Complex number 552 12.1.1 Rectangular form of complex number 552 12.1.2 Polar form of complex numbers 554 12.1.3 De Moivre’s theorem 557 12.2 Analytic functions and Cauchy-Riemann equation 562 12.2.1 Complex equation 562 12.2.2 Complex function 563 12.2.3 Derivative of a complex function, analytic functions 564 12.2.4 Cauchy-Riemann equations 565 12.2.5 Laplace’s equation and harmonic function 570 12.3 Several complex functions 574 12.3.1 Complex exponential functions and complex logarithms 574 12.3.2 Complex trigonometric functions and complex hyperbolic functions 576 13 Complex integration 582 13.1 Line integral in the complex plane 582 13.1.1 Complex line integral 582 13.1.2 Representation of parameters for a complex path 584

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xii Contents 13.1.3 Representation of parameters for a complex line integral 586 13.1.4 Upper bound of a complex line integral 591 13.2 Cauchy’s integral theorem I and II 595 13.3 Cauchy’s integral theorem III (derivative of an analytic function) 604 Bibliography 613 Appendix A: Basic formulas 614 Appendix B: MATLAB® 621 Index 647

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Preface Engineering mathematics is a foundational subject that all engineering stu- dents must study. Over the years, many excellent textbooks have been pub- lished to facilitate understanding. However, some of these books include more content than can be realistically covered in two semesters, and their verbose explanations often obscure the focus. Additionally, exercises in many textbooks are excessive in number and inconsistent in difculty, mak- ing it challenging to efciently master the concepts. This textbook addresses these challenges and aims to provide an engineer- ing mathematics resource tailored to the practical needs of university educa- tion. Its key features are as follows: 1. Clear and Concise Explanations: The core principles and concepts are explained succinctly and systemati- cally, with key ideas categorized under Remark for better organization. This approach helps students grasp the material more easily and enables instructors to teach more efectively. 2. Detailed Solutions: Each theoretical concept is accompanied by solved examples (Solution) that are thoroughly explained. To reinforce understanding, alternate so- lution methods (Another Solution) are provided, connecting the material to high school calculus or previously covered topics in the textbook. 3. Verifcation and Checkpoints: Complex calculations and the accuracy of answers are clarifed in desig- nated sections labeled as Check. 4. Integration with MATLAB®: Numerical analysis methods are included, and MATLAB® is utilized to simplify the problem-solving process. MATLAB®-based examples (M_ Example) are provided to deepen students’ understanding and practical skills. 5. Categorized Exercises: Problems are organized by example type, with additional applied prob- lems related to mechanical, electrical, electronic, and chemical engineer- ing included in each chapter.

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xiv Preface 6. Concept-Focused Problems: The exercises emphasize clarity and understanding rather than overly complicated calculations, allowing students to develop a frm grasp of concepts. 7. Optional Advanced Topics: Challenging chapters are marked as (*optional) and include detailed ex- planations for interested students. 8. Content Optimization for Two Semesters: The number of exercises has been adjusted to align with the realities of university coursework, making the material manageable within a two- semester timeframe. It is my hope that this textbook will become a valuable resource for en- gineering students. I extend my heartfelt gratitude to my son, Jaeyong, who reviewed and solved all the problems in this book, and to the team at CRC Press/Taylor & Francis Group, whose dedication made this publication possible. Chul Ki Song Main author MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: www.mathworks.com

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1 First-order ordinary diferential equations In the fundamental principles of classical mechanics established by Sir Isaac Newton, known as Newton’s laws, the position of an object is expressed as a function of time. By introducing the concepts of diferentiation, which yields velocity when applied to position and acceleration when applied to velocity, these laws enable the representation of physical concepts such as force, momentum, and energy through diferential equations. Therefore, diferential equations serve as the foundation for nearly all engineering and physics disciplines, including electrical and electronic engineering, structural mechanics, dynamics, thermodynamics, fuid mechanics, and aerospace engineering. 1.1 Diferential equations and their solutions 1.1.1 Fundamentals of diferential equations When we seek to interpret physical phenomena, we often mathematically formulate these phenomena using various variables and functions. The resulting mathematical expression is referred to as a mathematical model. Furthermore, this process of creating such equations is known as mathematical modeling. Most mathematical models involving physical concepts incorporate de- rivatives and are expressed as equations containing derivatives. These equa- tions are called diferential equations. By fnding solutions that satisfy these diferential equations and analyzing the characteristics of these solutions, we gain a deeper understanding of the physical properties involved. 1.1.2 First-order diferential equations In a diferential equation, if the highest order of the derivative involved is of the n-th degree, then that diferential equation is referred to as an n-th order (order n) diferential equation. In this chapter, we will focus on frst-order diferential equations. In other ˛ dy ̂ words, we will deal with equations that contain with y˜ = , ,y x, etc. ˝̇ dx ˇ̆ 1.1.3 Ordinary diferential equations and partial diferential equations An ordinary diferential equation (ODE) involves frst-order or higher-order derivatives. Examples of ODEs include the following: y˜ = sin x + 2, (1.1) y˜ + 2y˛ + 9 = −x , (1.2)y e y° − 3y = 0, ˜ ˜ (1.3) DOI: 10.1201/9781003608912-1

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2 Engineering Mathematics with MATLAB® where y is a function of the variable, which means = ( )y y x . Also, y″ 2 2 3 3and y‴ represent the second and third derivatives of d y dx , d y dx ,/ / respectively. On the other hand, a partial diferential equation (PDE) includes partial derivatives with respect to two or more variables. An example of PDEs is as follows: ˜2 u ˜2 u 2 + 2 = 0, (1.4) ˜ x ˜ y 4 2− x . where u is a function with two variables x and y, = ( , )u u x y . PDEs are extensively applied in engineering, but we will learn more about them in Chapter 11. 1.1.4 Representation of diferential equations In general, there are two methods for representing equations: implicit form and explicit form. For example, the equation of a circle with a radius of 2 can be expressed implicitly as x2 + y2 = 4, while it can be expressed explicitly as y = ± Therefore, frst-order ODEs can be represented in two diferent forms, as follows: F x( , ,y y˛) = 0, (1.5) y˛ = f (x y, ). (1.6) Here, Eq. (1.5) represents the implicit form, while Eq. (1.6) represents −2 2the explicit form. For example, x y˜ − 3y = 0 is an implicit form, while 2 2y˜ = 3x y is an explicit form. 1.1.5 Diferential equations and solutions In a system where a point moves around a circle with a radius of 2, the height y with respect to the angle of rotation x can be represented as follows: y = 2sin x. (1.7) Taking the derivative of this expression, we get y˜ = 2cos x. This derivative satisfes the following diferential equation: y˜2 + y2 = 4. (1.8) Therefore, Eq. (1.7) is called a solution of the diferential equation (1.8).

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  First-Order Ordinary Diferential Equations 3 Example 1.1 2 a. Prove that y = is a solution to the ODE xy˜ = −y. x b. Prove that = 2 (where C is arbitrary) is a solution to the ODEy Ce x y˜ = 2y. Solution a. Given that y˜ = − 2 ,2x Left-hand side (LHS): xy˜ = − 2 , x Right-hand side (RHS): − = −y 2 . x Therefore, LHS = RHS, demonstrating that y = 2 is indeed a solution to x the ODE xy˜ = −y. b. LHS: 2y˜ = 2Ce x , RHS: 22y = 2Ce x . Therefore, LHS = RHS, demonstrating that =y Ce2x is indeed a solution to the ODE y˜ = 2y. Example 1.2 Solve the following ODEs. a. y˜ = sin x + 2 1 b. y˜ = 1+ x2 1 c. y˜ = 1− x2 f x˛ ( ) d. y˛ = f x( ) Solution a. y = ˜ (sin x + 2)dx = − cos x + 2x + C . (Where C is the constant of integration.) Answer y = − cos x + 2 +x C

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    4 Engineering Mathematics with MATLAB® b. Substituting into x = tan˜ ˜( < ˆ ), we obtain dx = sec ˜ ˜2 2 d and 2 2 21+ x = + ˜ = sec ˜1 tan . Then, we get 1 1 2= 2 dx = 2 sec ° ° = ° + = x Cy d C arctan + ,˜ 1+ x ˜ sec ° where C is the constant of integration.  Answer y = arctanx + C c. By performing partial fraction decomposition, we can rewrite the expression as: 1 1 ̋ 1 1 ˇ = + .2 ˙̂ ̆1− x 2 1+ x 1− x Then, we get 1 ̋ 1 1 ˇdy = + dx. 2 ̇̂ 1+ x 1− x ̆ Therefore, 1 y = (ln 1+ x − ln 1− x ) + C 2 1 1+ x = ln + C. (C is the constant of integration.) 2 1− x 1 1+ x Answer y = ln + C 2 1− x f x˝ ( ) d. dy = dx f x( ) By integrating both sides, we obtain: y = ln f ( )x + C. (C is the constant of integration.) Answer y = ln f ( )x + C 1.1.6 Initial value problems Generally, the general solution of a diferential equation includes arbitrary con- stants, resulting in infnitely many solutions. To fnd a specifc solution, we use initial conditions. This type of diferential equation with the initial conditions is referred to as an initial value problem (IVP). In other words, an IVP is expressed in the following form: ˛ f ( ), ( ) = y0.y = x y , y x0 (1.9)

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First-Order Ordinary Diferential Equations 5 Example 1.3 Solve the following IVP: y˛ = 2 ,y y( )0 = 3. Solution In Example 1.1(b), we determined that the general solution for y˜ = 2y is y Ce  is the constant of integration. = 2x . (C ) (We will learn the direct method for this in Section 1.2.) From the solution = x and the initial condition y ( )0y Ce2 = 3, we obtain C = 3. Hence, the IVP has the solution y = 3e2x. This is the particular solution. Figure 1.1 shows the solution in Example 1.3. Figure 1.1 The solution in Example 1.3. 1.1.7 Diferentiation The following is a summary of fundamental diferentiation: • Diferentiation of a Rational Function d n n−1x = nx (a) dx