Differential Equations with Maple

I Maple Use and Programming.- 1 Introduction to Maple.- 1.1 Calculator or What?.- 1.1.1 First Contact.- 1.2 Programming in Maple.- 1.3 Maple Variables.- 1.3.1 Variable Types.- 1.3.2 Maple Expressions.- 1.3.3 Aggregate Types.- 1.3.4 Maple Tables.- 1.4 Maple Syntax.- 1.4.1 Function Declarations.- 1.4.2 Assignments.- 1.4.3 Conditionals.- 1.4.4 Loops.- 1.4.5 Maple Evaluation.- 1.4.6 Details about Maple Procedures.- 1.4.7 Procedure Arguments.- 1.4.8 Programming Gotchas.- 1.4.9 Maple Program Development.- II Differential Equations.- 2 Introduction to Differential Equations.- 2.1 Example Problems.- 2.1.1 Simple Rate Equations.- 2.1.2 Mechanical Problems.- 2.1.3 Electrical Models.- 2.1.4 Kinetics.- 2.1.5 Beam Models.- 2.2 Order, Dimension and Type.- 2.2.1 First and Higher Order.- 2.2.2 Partial Derivative Problems.- 2.3 Problems.- 3 First Order Equations.- 3.1 Separable Problems.- 3.1.1 Separable Examples.- 3.2 Level Sets and Exact Equations.- 3.2.1 Solving Exact Problems.- 3.2.2 Exact Equation Examples.- 3.3 Linear Equations.- 3.3.1 Linear Examples.- 3.4 Tricky Substitutions.- 3.4.1 Lagrange.- 3.4.2 Riccati.- 3.4.3 Mechanics.- 3.5 Do Solutions Exist?.- 3.5.1 The Issue.- 3.5.2 Existence Theorems.- 3.6 Problems.- 4 Introduction to Numerical Methods.- 4.1 The Game and Euler's Method.- 4.2 Solution Taylor Series.- 4.3 Runge-Kutta Methods.- 4.4 General Runge-Kutta Methods.- 4.5 Maple Numeric Routines.- 4.6 Calling all RK4's (and relatives).- 4.7 Variable Step Size Methods.- 4.7.1 Running RK4 With RK3.- 4.7.2 Adjusting h.- 4.8 Serious Methods.- 4.9 Problems.- 5 Higher Order Differential Equations.- 5.1 Equations of Order N.- 5.1.1 Vector Equations.- 5.1.2 Equivalent Formulations.- 5.2 Linear Independence and Wronskians.- 5.2.1 Linear Independence.- 5.2.2 Wronskians.- 5.3 Fundamental Solutions.- 5.3.1 Existence of Fundamental Solutions.- 5.3.2 Constructing Fundamental Solutions.- 5.4 General and Particular Solutions.- 5.5 Constant Coefficient Problems.- 5.5.1 Homogeneous Case.- 5.5.2 Undetermined Coefficients.- 5.6 Problems.- 6 Laplace Transform Methods.- 6.1 Basic Definition.- 6.1.1 Examples.- 6.2 New Wine From Old.- 6.2.1 Algebra and Tables.- 6.3 Maple Facilities.- 6.4 Derivatives and Laplace.- 6.5 High Order Problems by Laplace.- 6.5.1 Fundamental Solutions.- 6.5.2 Undetermined Coefficients Redux.- 6.6 Convolutions.- 6.6.1 Inhomogeneous Problems.- 6.7 Problems.- 7 Systems of Equations.- 7.1 Linear Systems.- 7.2 Bases and Eigenvectors.- 7.3 Diagonalization.- 7.3.1 Using Maple.- 7.4 Jordan Forms.- 7.4.1 Using Maple.- 7.5 Matrix Exponentials.- 7.5.1 Laplace Transforms.- 7.5.2 Using Maple.- 7.6 Problems.- 8 Stability.- 8.1 Second Order Problems.- 8.2 Harmonic Oscillator Again.- 8.3 Nodes and Spirals.- 8.4 Higher Order Problems.- 8.4.1 The Easy Cases.- 8.4.2 Jordan Form.- 8.5 Linearization.- 8.6 Introduction to Lyapunov Theory.- 8.6.1 Linear Problems.- 8.6.2 Applications.- 8.7 Problems.- 9 Periodic Problems.- 9.1 Periodic Inputs.- 9.2 Phasors.- 9.3 Fourier Series.- 9.4 Time Domain Methods.- 9.5 Periodic Coefficients.- 9.6 Fundamental Matrices.- 9.7 Stability.- 9.8 Floquet Representation.- 9.9 Problems.- 10 Impedances and Differential Equations.- 10.1 Introduction to Impedances.- 10.2 AC Circuits.- 10.3 Transient Circuits.- 10.4 Circuit Impedance Examples.- 10.5 Loops and Nodes.- 10.6 Problems.- 11 Partial Differential Equations.- 11.1 Basic Problems.- 11.1.1 Heat Equation.- 11.1.2 Wave Equation.- 11.1.3 Laplace's Equation.- 11.1.4 Beam Vibrations.- 11.2 Boundary Conditions.- 11.2.1 Heat and Diffusion Problems.- 11.2.2 Wave Equations.- 11.2.3 Laplace's Equation.- 11.3 Separation of Variables.- 11.3.1 Cooling of a Slab.- 11.3.2 Standing Wave Solutions.- 11.3.3 Steady State Heat Conduction.- 11.4 Problems.- III Maple Application Topics.- 12 Introduction to Maple Applications.- 13 Plotting With Maple.- 13.1 Maple Plotting Structures.- 13.2 Remember Tables.- 13.3 Plotting Numerical Results.- 13.4 Plotting Vector Variables.- 13.5 Further Plotting.- 14 Maple and Laplace Transforms.- 14.1 Walking Maple Through Problems.- 14.2 Coding a Beam Problem Solver.- 15 Maple Linear Algebra Applications.- 15.1 Canonical Forms.- 15.1.1 Eigenvalues and Eigenvectors.- 15.1.2 Jordan Form Calculations.- 15.2 Matrix Exponentials.- 15.3 Stability Examples.- 15.4 Periodic Solutions with Maple.- 16 Runge-Kutta Designs.- 16.1 Orientation and Overview.- 16.2 Notational Issues.- 16.3 True Solution Derivatives.- 16.4 Runge-Kutta Derivatives.- 16.5 Traps and Tricky Bits.- 16.6 A Sample Run.- 16.7 Order Condition Code.- 17 Maple Packages.- 17.1 Introduction.- 17.2 Constructing Packages.- 17.3 A Control Design Package.- 17.3.1 Root Locus.- 17.3.2 Nyquist Locus.- 17.3.3 Bode Plots.- 17.3.4 Classical vs. State Space.- 17.3.5 LQR Utility Code.- 17.3.6 Least Squares Optimal Control.- 17.3.7 Control Equation Generation.- 17.3.8 LTI Realization Calculations.- 17.3.9 Initialization.- 17.4 Package Creation and Installation.- 17.5 How About Help?.- 17.5.1 Maple Libraries.- 17.5.2 Creating and Installing Help Files.- 17.6 Control Package Demo.- IV Appendices.- A Review Problems.- B Laplace Transform Table.- References.