DOCSLIB.ORG

Second-Order Ordinary Differential Equation Solutions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Second-Order Ordinary Differential Equation Solutions 2nd Order Linear Ordinary Differential Equations Solutions for equations of the following general form: dy2 dy ++=ax12()axyhx () () dx2 dx Reduction of Order If terms are missing from the general second-order differential equation, it is sometimes possible to reduce the equation to a first-order ordinary differential equation. Second-order differential equations can be solved by reduction of order for two cases. Dependent Variable (y) is Missing dy2 dy +=ax1()hx () dx2 dx The procedures is to define a new variable p as: dy = p dx which can be differentiated again with respect to x to give: dy2 dp = dx2 dx These are substituted into the differential equation to give: dp +=axphx() () dx 1 which can then be solved by integrating factors to give (see handout on solution methods for 1st order differential equations): 1 ⎡ ⎤ p =+⎢ hxFxdx() () I1 ⎥ Fx()⎣∫ ⎦ ⎡ ⎤ where Fx()= exp ⎢ a1 () xdx⎥ ⎣∫ ⎦ The solution y is found by substituting for p = dy / dx and integrating again with respect to x. 1 ⎪⎪⎧⎫1 ⎡⎤ y =++⎨⎬⎢⎥hxFxdx() () I12 dx I ⎪⎪Fx()⎣⎦ ∫∫⎩⎭ ⎡⎤ where Fx()= exp ⎢⎥ a1 () xdx ⎣⎦∫ where I1 and I2 are constants of integration. (Note that throughout this document constants of integration will be indicated by this notation.) Independent Variable (x) is Missing dy2 dy ++=a12ay 0 dx2 dx The procedures is to define a new variable p as: dy = p dx which can be differentiated again with respect to x to give: dy2 dp = dx2 dx but remember that p can be written as a function of y which is a function of x. That is p = f(y(x)). This can be differentiated by chain rule to give (remembering that p = dy / dx): dp dp dy dp ==p dx dy dx dy These relationships for p are substituted into the differential equation to give: dp p ++=ap ay 0 dy 12 The 2nd order differential equation of y with respect to x has now been converted into a 1st order differential equation of p with respect to y. This equation is nonlinear (because p multiplies dp / dx) and can only be solved analytically if it is possible to separate and integrate. Once p is determined as a function of y (e.g., p = f(y)), then y can be found by integrating f(y) with respect to x to give: dy =+xI z fy() 2 The first constant of integration will be contained in the function f(y). 2 Variation of Parameters This method can be used anytime you already know one solution, yx1(), to the homogeneous form of the general differential equation given below. dy2 dy ++=ax12()axyhx () () dx2 dx The complete solution is found by substituting yuxyx= ()1 () into the above differential equation. The differentials of y are as follows: yuyuy′ = 11′ + ′ yuyuyuy′′ = 111′′+ 2 ′ ′ + ′′ which when substituted into the differential equation gives: uy′′ 111111121+ u′′bgbg2() y++ axy uy′′ + axy ()()()′ += a xy hx Since y1 is a solution to the homogeneous form of the differential equation shown at the top of the page, yaxyaxy11′′ + ( ) 12′ + ( ) 1= 0 , and the above equation reduces to give the following differential equation in u: uy′′1111++ u ′(2()() y ′ axy) = hx The function u can be found from this differential equation by reducing order and then solving by integrating factors. The complete solution y can be found by multiplying u by y1 to give the general solution: L 1 I O yuyy==hxyFdx() +2 dx+ I y 11M 2 1 2 P 11 zNM yF1 z yF1 QP where Fadxaˆ =≠exp⎡⎤ when 0 ⎣⎦∫ 11 ˆ Fa= 1 when 1 = 0 From which you can identify the second solution and the particular solution as follows: dx yy= 212 z yF1 1 L O yy= h() x y Fdx dx p 1 2 M 1 P z yF1 Nz Q 3 Constant Coefficient Ordinary Differential Equations dy2 dy a ++=b cy 0 dx2 dx where a, b and c are constants The form of the differential equation suggests solutions of ye= rx . (At this point this form is deduced by understanding the properties of differentiating erx . Later, we will develop a general approach for determining that this is the form of the solution.) yre′ = rx yre′′ = 2 rx ()ar2 ++ br c erx =0 −±bb2 −4 ac Characteristic Equation: r = 2a Case 1: Real & Unequal Roots (bac2 − 40> ) (a) If rr12≠ and rr12≠− , then rx12 rx yIe=+12 Ie (b) If rrr12=− = , then rx− rx yIeIe=+12 or yA= sinh( rxB )+ cosh( rx ) AB+ BA− where I = and I = 1 2 2 2 Case 2: Complex Roots (bac2 − 40< ) r = α ± iβ , complex conjugate 4 b 4ac− b2 where α=− and β= 2a 2a αβxxi-i β x ye=+e Ie12 Ie j But eix = cosx + i sin x αx yeIxixI=++−c 12cos(ββ ) sin( ) cos( ββ xix ) sin( ) h Let IAB111=+i and IAB222= + i and AA= 12+ A and BB=−21 B ye=+αx b Acos(ββ x ) B sin( x )g Case 3: Real and Equal Roots (bac2 − 40= ) b rrr===− 12 2a The characteristic equation gives only one solution, rx ye1 = The second solution can be found by variation of parameters to give: dx L b O F bxI yy21= where F = expdx = expG J 2 M a P H a K z yF1 Nz Q rx dx rx rx ye2 ===edxxe z ee−bx a bx a z rx rx yIeIxe=+12 5 Equidimensional Ordinary Differential Equation dy2 a dy b ++=y 0 dx22x dx x where a and b are constants The form of the differential equation suggests solutions of yx= r (At this point this form is deduced by understanding the properties of differentiating xr . Later, we will develop a general approach for determining that this is the form of the solution.) yrx′ = r−1 yrrx′′ =−()1 r−2 brr()−+10 ar + bxg r−2 = Characteristic Equation: rarb2 +−()10 += 114−±aab() −2 − r = 2 Case 1: Real & Unequal Roots ()140−−>ab2 rr12 yIxIx=+12 Case 2: Complex Roots ()140−−<ab2 r = α ± iβ 1− a 41ba22−−() where α= and β= 2 2 αβiii α−− β αβlnxx α i β ln yIxx=+12 Ixx = Ixe 1 + Ixe 2 But eix = cosx + i sin x Let IAB111=+i and IAB222= + i and AA= 12+ A and BB=−21 B αα yAx=+cos(ββ ln x ) Bx sin( ln x ) 6 Case 3: Real and Equal Roots ()140−−=ab2 1− a rrr=== 12 2 The characteristic equation gives only one solution, r yx1 = The second solution can be found by variation of parameters to give: dx L a O a yy21= where F = expdx== expbg a ln x x 2 M x P z yF1 Nz Q r dx r dx rrdx yx2 =====x x xxln z xx21()/−−aa 2z xx () 1 aa z x rr yIxIxx=+12ln 7 Special Equations Several second-order ordinary differential equations arise so often that they have been given names. Some of these are listed below. Harmonic Equation The following differential equation commonly arises for problems written in a rectangular coordinate system. dy2 +=by2 0 dx2 where b2 is not a function of x or y This differential equation is a constant coefficient equation with the solution: yI= 12sin( bxI )+ cos( bx ) Modified Harmonic Equation Like the harmonic equation, this equation commonly arises for problems written in a rectangular coordinate system. dy2 −=by2 0 dx2 where b2 is not a function of x or y This differential equation is a constant coefficient equation with the solution: yI= 12sinh( bxI )+ cosh( bx ) This equation can also be written in terms of the exponential as: yI= 12exp( bxI )+ exp(− bx ) As a general rule, it is usually convenient to use the sinh/cosh form of the solution for problems with finite boundaries and to use the exponential form of the solution for problems with one or more infinite boundaries. Bessel's Equation The following differential equation commonly arises for problems written in a cylindrical coordinate system. dy2 dy x2 ++x ejbx22 − p 2 y =0 dx2 dx 8 where b2 and p2 are constants. This differential equation has the solution yAJbxBJbx= pp()+ − () where A and B are constants of integration, and Jp is the Bessel function of the first kind and order p. If p is an integer or if p = 0, then the differential equation is: dy2 dy x2 ++x ebx22 − n 2j y =0 dx2 dx where n is an integer or zero. The solution to this equation is: yAJbxBYbx= nn()+ () where Yn is the Bessel function of the second kind and order n. The Bessel functions of the first and second kind are similar to the sine and cosine functions (i.e., solutions to the harmonic equation). In particular, like the sine and cosine functions, Bessel functions of the first and second kind are periodic for real arguments. Modified Bessel's Equation Like Bessel's equation, this equation commonly arises for problems written in cylindrical a coordinate system. dy2 dy x2 +−x ebx22 + p 2j y =0 dx2 dx where b2 and p2 are constants. This differential equation has the solution yAIbxBIbx= pp()+ − () where A and B are constants of integration, and Ip is the modified Bessel function of the first kind and order p. If p is an integer or if p = 0, then the differential equation is: dy2 dy x2 +−x ebx22 + n 2j y =0 dx2 dx where n is an integer or zero. The solution to this equation is: yAIbxBKbx= nn()+ () where Kn is the modified Bessel function of the second kind and order n. Modified Bessel functions of the first and second kind are similar to the hyperbolic sine and hyperbolic cosine functions (i.e., solutions to the modified harmonic equation).
Load more

Recommended publications
  • Mathematics 412 Partial Differential Equations
    Mathematics 412 Partial Differential Equations c S. A. Fulling Fall, 2005 The Wave Equation This introductory example will have three parts.* 1. I will show how a particular, simple partial differential equation (PDE) arises in a physical problem. 2. We’ll look at its solutions, which happen to be unusually easy to find in this case. 3. We’ll solve the equation again by separation of variables, the central theme of this course, and see how Fourier series arise. The wave equation in two variables (one space, one time) is ∂2u ∂2u = c2 , ∂t2 ∂x2 where c is a constant, which turns out to be the speed of the waves described by the equation. Most textbooks derive the wave equation for a vibrating string (e.g., Haber- man, Chap. 4). It arises in many other contexts — for example, light waves (the electromagnetic field). For variety, I shall look at the case of sound waves (motion in a gas). Sound waves Reference: Feynman Lectures in Physics, Vol. 1, Chap. 47. We assume that the gas moves back and forth in one dimension only (the x direction). If there is no sound, then each bit of gas is at rest at some place (x,y,z). There is a uniform equilibrium density ρ0 (mass per unit volume) and pressure P0 (force per unit area). Now suppose the gas moves; all gas in the layer at x moves the same distance, X(x), but gas in other layers move by different distances. More precisely, at each time t the layer originally at x is displaced to x + X(x,t).
    [Show full text]
  • Chapter 9 Some Special Functions
    Chapter 9 Some Special Functions Up to this point we have focused on the general properties that are associated with uniform convergence of sequences and series of functions. In this chapter, most of our attention will focus on series that are formed from sequences of functions that are polynomials having one and only one zero of increasing order. In a sense, these are series of functions that are “about as good as it gets.” It would be even better if we were doing this discussion in the “Complex World” however, we will restrict ourselves mostly to power series in the reals. 9.1 Power Series Over the Reals jIn this section,k we turn to series that are generated by sequences of functions :k * ck x k0. De¿nition 9.1.1 A power series in U about the point : + U isaseriesintheform ;* n c0 cn x : n1 where : and cn, for n + M C 0 , are real constants. Remark 9.1.2 When we discuss power series, we are still interested in the differ- ent types of convergence that were discussed in the last chapter namely, point- 3wise, uniform and absolute. In this context, for example, the power series c0 * :n t U n1 cn x is said to be3pointwise convergent on a set S 3if and only if, + * :n * :n for each x0 S, the series c0 n1 cn x0 converges. If c0 n1 cn x0 369 370 CHAPTER 9. SOME SPECIAL FUNCTIONS 3 * :n is divergent, then the power series c0 n1 cn x is said to diverge at the point x0.
    [Show full text]
  • Modifications of the Method of Variation of Parameters
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector International Joumal Available online at www.sciencedirect.com computers & oc,-.c~ ~°..-cT- mathematics with applications Computers and Mathematics with Applications 51 (2006) 451-466 www.elsevier.com/locate/camwa Modifications of the Method of Variation of Parameters ROBERTO BARRIO* GME, Depto. Matem~tica Aplicada Facultad de Ciencias Universidad de Zaragoza E-50009 Zaragoza, Spain rbarrio@unizar, es SERGIO SERRANO GME, Depto. MatemAtica Aplicada Centro Politdcnico Superior Universidad de Zaragoza Maria de Luna 3, E-50015 Zaragoza, Spain sserrano©unizar, es (Received February PO05; revised and accepted October 2005) Abstract--ln spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. ~) 2006 Elsevier Ltd. All rights reserved. Keywords--Variation of parameters, Lagrange equations, Gauss equations, Satellite orbits, Dif- ferential equations. 1. INTRODUCTION The method of variation of parameters was developed by Leonhard Euler in the middle of XVIII century to describe the mutual perturbations of Jupiter and Saturn. However, his results were not absolutely correct because he did not consider the orbital elements varying simultaneously. In 1766, Lagrange improved the method developed by Euler, although he kept considering some of the orbital elements as constants, which caused some of his equations to be incorrect.
    [Show full text]
  • Special Functions
    Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS table of content Chapter 5 Special Functions 5.1 Heaviside step function - filter function 5.2 Dirac delta function - modeling of impulse processes 5.3 Sine integral function - Gibbs phenomena 5.4 Error function 5.5 Gamma function 5.6 Bessel functions 1. Bessel equation of order ν (BE) 2. Singular points. Frobenius method 3. Indicial equation 4. First solution – Bessel function of the 1st kind 5. Second solution – Bessel function of the 2nd kind. General solution of Bessel equation 6. Bessel functions of half orders – spherical Bessel functions 7. Bessel function of the complex variable – Bessel function of the 3rd kind (Hankel functions) 8. Properties of Bessel functions: - oscillations - identities - differentiation - integration - addition theorem 9. Generating functions 10. Modified Bessel equation (MBE) - modified Bessel functions of the 1st and the 2nd kind 11. Equations solvable in terms of Bessel functions - Airy equation, Airy functions 12. Orthogonality of Bessel functions - self-adjoint form of Bessel equation - orthogonal sets in circular domain - orthogonal sets in annular fomain - Fourier-Bessel series 5.7 Legendre Functions 1. Legendre equation 2. Solution of Legendre equation – Legendre polynomials 3. Recurrence and Rodrigues’ formulae 4. Orthogonality of Legendre polynomials 5. Fourier-Legendre series 6. Integral transform 5.8 Exercises Chapter 5 SPECIAL FUNCTIONS Chapter 5 SPECIAL FUNCTIONS Introduction In this chapter we summarize information about several functions which are widely used for mathematical modeling in engineering. Some of them play a supplemental role, while the others, such as the Bessel and Legendre functions, are of primary importance. These functions appear as solutions of boundary value problems in physics and engineering.
    [Show full text]
  • Notes on Special Functions
    Spring 2005 1 Notes on Special Functions Francis J. Narcowich Department of Mathematics Texas A&M University College Station, TX 77843-3368 Introduction These notes are for our classes on special functions. The elementary functions that appear in the first few semesters of calculus – powers of x, ln, sin, cos, exp, etc. are not enough to deal with the many problems that arise in science and engineering. Special function is a term loosely applied to additional functions that arise frequently in applications. We will discuss three of them here: Bessel functions, the gamma function, and Legendre polynomials. 1 Bessel Functions 1.1 A heat flow problem Bessel functions come up in problems with circular or spherical symmetry. As an example, we will look at the problem of heat flow in a circular plate. This problem is the same as the one for diffusion in a plate. By adjusting our units of length and time, we may assume that the plate has a unit radius. Also, we may assume that certain physical constants involved are 1. In polar coordinates, the equation governing the temperature at any point (r, θ) and any time t is the heat equation, ∂u 1 ∂ ∂u 1 ∂2u = r + 2 2 . (1) ∂t r ∂r³ ∂r ´ r ∂θ By itself, this equation only represents the energy balance in the plate. Two more pieces of information are necessary for us to be able to determine the temperature: the initial temperature, u(0, r, θ)= f(r, θ), and the temperature at the boundary, u(t, 1, θ), which we will assume to be 0.
    [Show full text]
  • The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ....................................
    [Show full text]
  • MTHE / MATH 237 Differential Equations for Engineering Science
    i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations ..................................................
    [Show full text]
  • [AVV5] We Studied Classical Special Functions, Such As the Gamma Function, the Hypergeometric Function 2F1, and the Complete Elliptic Inte- Grals
    CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 11, Pages 250–270 (November 8, 2007) S 1088-4173(07)00168-3 TOPICS IN SPECIAL FUNCTIONS. II G. D. ANDERSON, M. K. VAMANAMURTHY, AND M. VUORINEN Dedicated to Seppo Rickman and Jussi V¨ais¨al¨a. Abstract. In geometric function theory, conformally invariant extremal prob- lems often have expressions in terms of special functions. Such problems occur, for instance, in the study of change of euclidean and noneuclidean distances under quasiconformal mappings. This fact has led to many new results on special functions. Our goal is to provide a survey of such results. 1. Introduction In our earlier survey [AVV5] we studied classical special functions, such as the gamma function, the hypergeometric function 2F1, and the complete elliptic inte- grals. In this survey our focus is different: We will concentrate on special functions that occur in the distortion theory of quasiconformal maps. The definitions of these functions involve the aforementioned functions, and therefore these two surveys are closely connected. However, we hope to minimize overlap with the earlier survey. Extremal problems of geometric function theory lead naturally to the study of special functions. L. V. Ahlfors’s book [Ah] on quasiconformal mappings, where the extremal length of a curve family is a crucial tool, discusses among other things the extremal problems of Gr¨otzsch and Teichm¨uller and provides methods for studying the boundary correspondence under quasiconformal automorphisms of the upper half plane or the unit disk. These problems have led to many ramifications and have greatly stimulated later research.
    [Show full text]
  • Partial Differential Equations: an Introduction with Mathematica And
    PARTIAL DIFFERENTIAL EQUATIONS (Second Edition) An Introduction with Mathernatica and MAPLE This page intentionally left blank PARTIAL DIFFERENTIAL EQUATIONS (Scond Edition) An Introduction with Mathematica and MAPLE Ioannis P Stavroulakis University of Ioannina, Greece Stepan A Tersian University of Rozousse, Bulgaria WeWorld Scientific NEW JERSEY 6 LONDON * SINGAPORE * BElJlNG SHANGHAI * HONG KONG * TAIPEI CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PARTIAL DIFFERENTIAL EQUATIONS An Introduction with Mathematica and Maple (Second Edition) Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book or parts thereoj may not be reproduced in anyform or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-238-815-X Printed in Singapore. To our wives Georgia and Mariam and our children Petros, Maria-Christina and Ioannis and Takuhi and Lusina This page intentionally left blank Preface In this second edition the section “Weak Derivatives and Weak Solutions” was removed to Chapter 5 to be together with advanced concepts such as discontinuous solutions of nonlinear conservation laws.
    [Show full text]
  • Math 6730 : Asymptotic and Perturbation Methods Hyunjoong
    Math 6730 : Asymptotic and Perturbation Methods Hyunjoong Kim and Chee-Han Tan Last modified : January 13, 2018 2 Contents Preface 5 1 Introduction to Asymptotic Approximation7 1.1 Asymptotic Expansion................................8 1.1.1 Order symbols.................................9 1.1.2 Accuracy vs convergence........................... 10 1.1.3 Manipulating asymptotic expansions.................... 10 1.2 Algebraic and Transcendental Equations...................... 11 1.2.1 Singular quadratic equation......................... 11 1.2.2 Exponential equation............................. 14 1.2.3 Trigonometric equation............................ 15 1.3 Differential Equations: Regular Perturbation Theory............... 16 1.3.1 Projectile motion............................... 16 1.3.2 Nonlinear potential problem......................... 17 1.3.3 Fredholm alternative............................. 19 1.4 Problems........................................ 20 2 Matched Asymptotic Expansions 31 2.1 Introductory example................................. 31 2.1.1 Outer solution by regular perturbation................... 31 2.1.2 Boundary layer................................ 32 2.1.3 Matching................................... 33 2.1.4 Composite expression............................. 33 2.2 Extensions: multiple boundary layers, etc...................... 34 2.2.1 Multiple boundary layers........................... 34 2.2.2 Interior layers................................. 35 2.3 Partial differential equations............................. 36 2.4 Strongly
    [Show full text]
  • ON MEANS, POLYNOMIALS and SPECIAL FUNCTIONS Johan Gielis
    THE TEACHING OF MATHEMATICS 2014, Vol. XVII, 1, pp. 1–20 ON MEANS, POLYNOMIALS AND SPECIAL FUNCTIONS Johan Gielis, Rik Verhulst, Diego Caratelli, Paolo E. Ricci, and Ilia Tavkhelidze Abstract. We discuss how derivatives can be considered as a game of cubes and beams, and of geometric means. The same principles underlie wide classes of polynomials. This results in an unconventional view on the history of the differentiation and differentials. MathEduc Subject Classification: F50, H20 MSC Subject Classification: 97E55, 97H25, 51M15 Key words and phrases: Derivative; polynomial; special function; arithmetic, geometric and harmonic means. 1. On derivation and derivatives In On Proof and Progress in Mathematics William Thurston describes how people develop an “understanding” of mathematics [31]. He uses the example of derivatives, that can be approached from very different viewpoints, allowing individuals to develop their own understanding of derivatives. “The derivative can be thought of as: (1) Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function. (2) Symbolic: the derivative of xn is nxn¡1, the derivative of sin (x) is cos (x), the derivative of f ± g is f 0 ± g £ g0, etc ::: (3) Logical: f 0 (x) = d if and only if for every ² there is a ±, such that 0 < j∆xj < ± when ¯ ¯ ¯f(x + ∆x) ¡ f(x) ¯ ¯ ¡ d¯ < ± 1 ¯ ∆x ¯ (4) Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent. (5) Rate: the instantaneous speed of f (t) = d, when t is time.
    [Show full text]
  • Modified Variation of Parameters Method for Differential Equations
    World Applied Sciences Journal 6 (10): 1372-1376, 2009 ISSN 1818-4952 © IDOSI Publications, 2009 Modified Variation of Parameters Method for Differential Equations 1Syed Tauseef Mohyud-Din, 2Muhammad Aslam Noor and 2Khalida Inayat Noor 1HITEC University, Taxila Cantt, Pakistan 2Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract: In this paper, we apply the Modified Variation of Parameters Method (MVPM) for solving nonlinear differential equations which are associated with oscillators. The proposed modification is made by the elegant coupling of traditional Variation of Parameters Method (VPM) and He’s polynomials. The suggested algorithm is more efficient and easier to handle as compare to decomposition method. Numerical results show the efficiency of the proposed algorithm. Key words: Variation of parameters method • He’s polynomials • nonlinear oscillator INTRODUCTION the inbuilt deficiencies of various existing techniques. Numerical results show the complete reliability of the The nonlinear oscillators appear in various proposed technique. physical phenomena related to physics, applied and engineering sciences [1-6] and the references therein. VARIATION OF Several techniques including variational iteration, PARAMETERS METHOD (VPM) homotopy perturbation and expansion of parameters have been applied for solving such problems [1-6]. Consider the following second-order partial He [2-9] developed the homotopy perturbation differential equation: method for solving various physical problems. This reliable technique has been applied to a wide range of ytt = f(t,x,y,z,y,y,y,yxyzxx ,yyy ,y)zz (1) diversified physical problems [1-23] and the references therein. Recently, Ghorbani et al. [10, 11] introduced He’s polynomials by splitting the nonlinear term into a where t such that (-∞<t<∞) is time and f is linear or non series of polynomials.
    [Show full text]