CONTENTS vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory-Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362CONTENTS ● vii 7 THE LAPLACE TRANSFORM 255 7.1 Definition of the Laplace Transform 256 7.2 Inverse Transforms and Transforms of Derivatives 262 7.2.1 Inverse Transforms 262 7.2.2 Transforms of Derivatives 265 7.3 Operational Properties I 270 7.3.1 Translation on the s-Axis 271 7.3.2 Translation on the t-Axis 274 7.4 Operational Properties II 282 7.4.1 Derivatives of a Transform 282 7.4.2 Transforms of Integrals 283 7.4.3 Transform of a Periodic Function 287 7.5 The Dirac Delta Function 292 7.6 Systems of Linear Differential Equations 295 CHAPTER 7 IN REVIEW 300 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 303 8.1 Preliminary Theory—Linear Systems 304 8.2 Homogeneous Linear Systems 311 8.2.1 Distinct Real Eigenvalues 312 8.2.2 Repeated Eigenvalues 315 8.2.3 Complex Eigenvalues 320 8.3 Nonhomogeneous Linear Systems 326 8.3.1 Undetermined Coefficients 326 8.3.2 Variation of Parameters 329 8.4 Matrix Exponential 334 CHAPTER 8 IN REVIEW 337 9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 339 9.1 Euler Methods and Error Analysis 340 9.2 Runge-Kutta Methods 345 9.3 Multistep Methods 350 9.4 Higher-Order Equations and Systems 353 9.5 Second-Order Boundary-Value Problems 358 CHAPTER 9 IN REVIEW 362