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ORDINARY DIFFERENTIAL EQUATIONS LAPLACE TRANSFORMS AND NUMERICAL METHODS FOR ENGINEERS by Steven J. DESJARDINS and R´emi VAILLANCOURT Notes for the course MAT 2384 3X Spring 2011 D´epartement de math´ematiques et de statistique Department of Mathematics and Statistics Universit´e d’Ottawa / University of Ottawa Ottawa, ON, Canada K1N 6N5 2011.04.01 i

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ii DESJARDINS, Steven J. Department of Mathematics and Statistics University of Ottawa Ottawa, Ontario, Canada K1N 6N5 e-mail: [email protected] homepage: http://www.mathstat.uottawa.ca/~sdesj740 VAILLANCOURT, R´emi D´epartement de math´ematiques et de statistique Universit´e d’Ottawa Ottawa (Ontario), Canada K1N 6N5 courriel: [email protected] page d’accueil: http://www.site.uottawa.ca/~remi The production of this book beneﬁtted from grants from the Natural Sciences and Engineering Research Council of Canada. ⃝c S. J. Desjardins and R. Vaillancourt, Ottawa 2011

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Contents Part 1. Diﬀerential Equations and Laplace Transforms 1 Chapter 1. First-Order Ordinary Diﬀerential Equations 3 1.1. Fundamental Concepts 3 1.2. Separable Equations 5 1.3. Equations with Homogeneous Coeﬃcients 7 1.4. Exact Equations 9 1.5. Integrating Factors 16 1.6. First-Order Linear Equations 21 1.7. Orthogonal Families of Curves 23 1.8. Direction Fields and Approximate Solutions 26 1.9. Existence and Uniqueness of Solutions 26 Chapter 2. Second-Order Ordinary Diﬀerential Equations 33 2.1. Linear Homogeneous Equations 33 2.2. Homogeneous Equations with Constant Coeﬃcients 33 2.3. Basis of the Solution Space 34 2.4. Independent Solutions 36 2.5. Modeling in Mechanics 39 2.6. Euler–Cauchy Equations 44 Chapter 3. Linear Diﬀerential Equations of Arbitrary Order 49 3.1. Homogeneous Equations 49 3.2. Linear Homogeneous Equations 55 3.3. Linear Nonhomogeneous Equations 59 3.4. Method of Undetermined Coeﬃcients 61 3.5. Particular Solution by Variation of Parameters 65 3.6. Forced Oscillations 71 Chapter 4. Systems of Diﬀerential Equations 77 4.1. Introduction 77 4.2. Existence and Uniqueness Theorem 79 4.3. Fundamental Systems 80 4.4. Homogeneous Linear Systems with Constant Coeﬃcients 83 4.5. Nonhomogeneous Linear Systems 91 Chapter 5. Laplace Transform 97 5.1. Deﬁnition 97 5.2. Transforms of Derivatives and Integrals 102 5.3. Shifts in s and in t 106 5.4. Dirac Delta Function 115 iii

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iv CONTENTS 5.5. Derivatives and Integrals of Transformed Functions 117 5.6. Laguerre Diﬀerential Equation 120 5.7. Convolution 122 5.8. Partial Fractions 125 5.9. Transform of Periodic Functions 125 Chapter 6. Power Series Solutions 129 6.1. The Method 129 6.2. Foundation of the Power Series Method 131 6.3. Legendre Equation and Legendre Polynomials 139 6.4. Orthogonality Relations for Pn(x) 142 6.5. Fourier–Legendre Series 145 6.6. Derivation of Gaussian Quadratures 148 Part 2. Numerical Methods 153 Chapter 7. Solutions of Nonlinear Equations 155 7.1. Computer Arithmetic 155 7.2. Review of Calculus 158 7.3. The Bisection Method 158 7.4. Fixed Point Iteration 162 7.5. Newton’s, Secant, and False Position Methods 167 7.6. Aitken–Steﬀensen Accelerated Convergence 175 7.7. Horner’s Method and the Synthetic Division 177 7.8. Mu¨ller’s Method 179 Chapter 8. Interpolation and Extrapolation 183 8.1. Lagrange Interpolating Polynomial 183 8.2. Newton’s Divided Diﬀerence Interpolating Polynomial 185 8.3. Gregory–Newton Forward-Diﬀerence Polynomial 189 8.4. Gregory–Newton Backward-Diﬀerence Polynomial 191 8.5. Hermite Interpolating Polynomial 192 8.6. Cubic Spline Interpolation 194 Chapter 9. Numerical Diﬀerentiation and Integration 197 9.1. Numerical Diﬀerentiation 197 9.2. The Eﬀect of Roundoﬀ and Truncation Errors 199 9.3. Richardson’s Extrapolation 201 9.4. Basic Numerical Integration Rules 203 9.5. The Composite Midpoint Rule 206 9.6. The Composite Trapezoidal Rule 208 9.7. The Composite Simpson Rule 210 9.8. Romberg Integration for the Trapezoidal Rule 212 9.9. Adaptive Quadrature Methods 213 9.10. Gaussian Quadrature 215 Chapter 10. Numerical Solution of Diﬀerential Equations 217 10.1. Initial Value Problems 217 10.2. Euler’s and Improved Euler’s Methods 218 10.3. Low-Order Explicit Runge–Kutta Methods 221

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CONTENTS v 10.4. Convergence of Numerical Methods 229 10.5. Absolutely Stable Numerical Methods 230 10.6. Stability of Runge–Kutta Methods 231 10.7. Embedded Pairs of Runge–Kutta Methods 234 10.8. Multistep Predictor-Corrector Methods 240 10.9. Stiﬀ Systems of Diﬀerential Equations 252 Part 3. Exercises and Solutions 261 Chapter 11. Exercises for Diﬀerential Equations and Laplace Transforms 263 Exercises for Chapter 1 263 Exercises for Chapter 2 265 Exercises for Chapter 3 266 Exercises for Chapter 4 268 Exercises for Chapter 5 269 Exercises for Chapter 6 271 Chapter 12. Exercises for Numerical Methods 275 Exercises for Chapter 7 275 Exercises for Chapter 8 277 Exercises for Chapter 9 278 Exercises for Chapter 10 280 Solutions to Starred Exercises 283 Solutions to Exercises from Chapters 1 to 6 283 Solutions to Exercises from Chapter 7 292 Solutions to Exercises for Chapter 8 294 Solutions to Exercises for Chapter 10 295 Part 4. Formulas and Tables 301 Chapter 13. Formulas and Tables 303 13.1. Integrating Factor of M(x, y) dx + N(x, y) dy = 0 303 13.2. Solution of First-Order Linear Diﬀerential Equations 303 13.3. Laguerre Polynomials on 0 ≤ x < ∞ 303 13.4. Legendre Polynomials Pn(x) on [−1, 1] 304 13.5. Fourier–Legendre Series Expansion 305 13.6. Table of Integrals 306 13.7. Table of Laplace Transforms 306

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Part 1 Diﬀerential Equations and Laplace Transforms

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CHAPTER 1 First-Order Ordinary Diﬀerential Equations 1.1. Fundamental Concepts (a) A diﬀerential equation is an equation involving an unkonwn function y, derivatives of it and functions of the independent variable. ′ d Here are three ordinary diﬀerential equations, where := : dx ′ (1) y = cos x, ′′ (2) y + 4y = 0, 2 ′′′ ′ x ′′ 2 2 (3) x y y + 2 e y = (x + 2)y . Here is a partial diﬀerential equation: 2 2 ∂ u ∂ u + = 0. 2 2 ∂x ∂y (b) The order of a diﬀerential equation is equal to the highest-order derivative that appears in it. The above equations (1), (2) and (3) are of order 1, 2 and 3, respectively. (c) An explicit solution of a diﬀerential equation with independent variable x on ]a, b[ is a function y = g(x) of x such that the diﬀerential equation becomes ′ ′ an identity in x on ]a, b[ when g(x), g (x), etc. are substituted for y, y , etc. in the diﬀerential equation. The solution y = g(x) describes a curve, or trajectory, in the xy-plane. We see that the function 2x y(x) = e is an explicit solution of the diﬀerential equation dy = 2y. dx In fact, we have ′ 2x L.H.S. := y (x) = 2 e , 2x R.H.S. := 2y(x) = 2 e . Hence L.H.S. = R.H.S., for all x. We thus have an identity in x on ] −∞,∞[. □ 3

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4 1. FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS (d) An implicit solution of a diﬀerential equation is a curve which is deﬁned by an equation of the form G(x, y) = c where c is an arbitrary constant. Note that G(x, y) represents a surface, a 2-dimensional object in 3-dimensional space where x and y are independent variables. By setting G(x, y) = c, a rela- tionship is created between x and y. We remark that an implicit solution always contains an equal sign, “=”, followed by a constant, otherwise z = G(x, y) represents a surface and not a curve. We see that the curve in the xy-plane, 2 2 x + y − 1 = 0, y > 0, is an implicit solution of the diﬀerential equation ′ yy = −x, on − 1 < x < 1. In fact, letting y be a function of x and diﬀerentiating the equation of the curve with respect to x, d d 2 2 (x + y − 1) = (0) = 0, dx dx we obtain ′ ′ 2x + 2yy = 0 or yy = −x. □ (e) The general solution of a diﬀerential equation of order n contains n arbi- trary constants. The one-parameter family of functions y(x) = sin x + c is the general solution of the ﬁrst-order diﬀerential equation ′ y (x) = cos x. This inﬁnite family of curves all have the same slope, and hence all members of this familiy are solutions of the diﬀerential equation. The general solution is written y(x) = sin x + c (with the arbitrary constant) to represent all of the possible solutions. Putting c = 1, we have the unique solution, y(x) = sinx + 1, 2 which goes through the point (0, 1) of R . Given an arbitrary point (x0, y0) of the plane, there is one and only one curve of the family which goes through that point. (See Fig. 1.1(a)). Similarly, we see that the one-parameter family of functions x y(x) = c e is the general solution of the diﬀerential equation ′ y = y. Setting c = −1, we have the unique solution, x y(x) = −e ,

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1.2. SEPARABLE EQUATIONS 5 y y c = 1� c = 1� 1 1 � � c = 0� 0 � x 0 x –1 c = – 1� –2 c = –2� –2 � � c = –2� (a) (b) � Figure 1.1. (a) Two one-parameter families of curves: (a) y = sin x + c; (b) y(x) = c exp(x). 2 which goes through the point (0,−1) of R . Given an arbitrary point (x0, y0) of the plane, there is one and only one curve of the family which goes through that point. (See Fig. 1.1(b)). 1.2. Separable Equations A diﬀerential equation is called separable if it can be written in the form dy g(y) = f(x). (1.1) dx We rewrite the equation using the diﬀerentials dy and dx and separate it by grouping on the left-hand side all terms containing y and on the right-hand side all terms containing x: g(y) dy = f(x) dx. (1.2) The solution of a separated equation is obtained by taking the indeﬁnite integral (primitive or antiderivative) of both sides and adding an arbitrary constant: ∫ ∫ g(y) dy = f(x) dx + c, (1.3) that is G(y) = F(x) + c. Only one constant is needed and it is placed on the right-hand side (i.e. on the side with the independent variable). The two forms of the implicit solution, G(y) = F(x) + c, or K(x, y) = −F(x) + G(y) = c, deﬁne y as a function of x or x as a function of y. Letting y = y(x) be a function of x, we verify that (1.3) is a solution of (1.1): d d ′ ′ ′ (LHS) = G (y(x)) = G (y(x)) y (x) = g(y)y , dx dx d d ′ (RHS) = [F(x) + c] = F (x) = f(x). □ dx dx ′ 2 Example 1.1. Solve y = 1 + y .

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