i ·CONTENTS 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory-Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients-Superposition Approach 140 4.5 Undetermined Coefficients-Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models:Initial-Value Problems 182 5.1.1 Spring/Mass Systems:Free Undamped Motion 182 5.1.2 Spring/Mass Systems:Free Damped Motion 186 5.1.3 Spring/Mass Systems:Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models:Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel's Equation 241 6.3.2 Legendre's Equation 248 CHAPTER 6 IN REVIEW 2535 4 vi ● CONTENTS HIGHER-ORDER DIFFERENTIAL EQUATIONS 117 4.1 Preliminary Theory—Linear Equations 118 4.1.1 Initial-Value and Boundary-Value Problems 118 4.1.2 Homogeneous Equations 120 4.1.3 Nonhomogeneous Equations 125 4.2 Reduction of Order 130 4.3 Homogeneous Linear Equations with Constant Coefficients 133 4.4 Undetermined Coefficients—Superposition Approach 140 4.5 Undetermined Coefficients—Annihilator Approach 150 4.6 Variation of Parameters 157 4.7 Cauchy-Euler Equation 162 4.8 Solving Systems of Linear DEs by Elimination 169 4.9 Nonlinear Differential Equations 174 CHAPTER 4 IN REVIEW 178 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 181 5.1 Linear Models: Initial-Value Problems 182 5.1.1 Spring/Mass Systems: Free Undamped Motion 182 5.1.2 Spring/Mass Systems: Free Damped Motion 186 5.1.3 Spring/Mass Systems: Driven Motion 189 5.1.4 Series Circuit Analogue 192 5.2 Linear Models: Boundary-Value Problems 199 5.3 Nonlinear Models 207 CHAPTER 5 IN REVIEW 216 SERIES SOLUTIONS OF LINEAR EQUATIONS 219 6.1 Solutions About Ordinary Points 220 6.1.1 Review of Power Series 220 6.1.2 Power Series Solutions 223 6.2 Solutions About Singular Points 231 6.3 Special Functions 241 6.3.1 Bessel’s Equation 241 6.3.2 Legendre’s Equation 248 CHAPTER 6 IN REVIEW 253 6