Student Solutions Manual for Fundamentals of Differential Equations and Fundamentals of Differential Equations and Boundary Value Problems

1. Introduction
   • 1.1 Background
   • 1.2 Solutions and Initial Value Problems
   • 1.3 Direction Fields
   • 1.4 The Approximation Method of Euler
2. First-Order Differential Equations
   • 2.1 Introduction: Motion of a Falling Body
   • 2.2 Separable Equations
   • 2.3 Linear Equations
   • 2.4 Exact Equations
   • 2.5 Special Integrating Factors
   • 2.6 Substitutions and Transformations
3. Mathematical Models and Numerical Methods Involving First Order Equations
   • 3.1 Mathematical Modeling
   • 3.2 Compartmental Analysis
   • 3.3 Heating and Cooling of Buildings
   • 3.4 Newtonian Mechanics
   • 3.5 Electrical Circuits
   • 3.6 Improved Euler’s Method
   • 3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta
4. Linear Second-Order Equations
   • 4.1 Introduction: The Mass-Spring Oscillator
   • 4.2 Homogeneous Linear Equations: The General Solution
   • 4.3 Auxiliary Equations with Complex Roots
   • 4.4 Nonhomogeneous Equations: The Method of Undetermined Coefficients
   • 4.5 The Superposition Principle and Undetermined Coefficients Revisited
   • 4.6 Variation of Parameters
   • 4.7 Variable-Coefficient Equations
   • 4.8 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations
   • 4.9 A Closer Look at Free Mechanical Vibrations
   • 4.10 A Closer Look at Forced Mechanical Vibrations
5. Introduction to Systems and Phase Plane Analysis
   • 5.1 Interconnected Fluid Tanks
   • 5.2 Elimination Method for Systems with Constant Coefficients
   • 5.3 Solving Systems and Higher-Order Equations Numerically
   • 5.4 Introduction to the Phase Plane
   • 5.5 Applications to Biomathematics: Epidemic and Tumor Growth Models
   • 5.6 Coupled Mass-Spring Systems
   • 5.7 Electrical Systems
   • 5.8 Dynamical Systems, Poincaré Maps, and Chaos
6. Theory of Higher-Order Linear Differential Equations
   • 6.1 Basic Theory of Linear Differential Equations
   • 6.2 Homogeneous Linear Equations with Constant Coefficients
   • 6.3 Undetermined Coefficients and the Annihilator Method
   • 6.4 Method of Variation of Parameters
7. Laplace Transforms
   • 7.1 Introduction: A Mixing Problem
   • 7.2 Definition of the Laplace Transform
   • 7.3 Properties of the Laplace Transform
   • 7.4 Inverse Laplace Transform
   • 7.5 Solving Initial Value Problems
   • 7.6 Transforms of Discontinuous Functions
   • 7.7 Transforms of Periodic and Power Functions
   • 7.8 Convolution
   • 7.9 Impulses and the Dirac Delta Function
   • 7.10 Solving Linear Systems with Laplace Transforms
8. Series Solutions of Differential Equations
   • 8.1 Introduction: The Taylor Polynomial Approximation
   • 8.2 Power Series and Analytic Functions
   • 8.3 Power Series Solutions to Linear Differential Equations
   • 8.4 Equations with Analytic Coefficients
   • 8.5 Cauchy-Euler (Equidimensional) Equations
   • 8.6 Method of Frobenius
   • 8.7 Finding a Second Linearly Independent Solution
   • 8.8 Special Functions
9. Matrix Methods for Linear Systems
   • 9.1 Introduction
   • 9.2 Review 1: Linear Algebraic Equations
   • 9.3 Review 2: Matrices and Vectors
   • 9.4 Linear Systems in Normal Form
   • 9.5 Homogeneous Linear Systems with Constant Coefficients
   • 9.6 Complex Eigenvalues
   • 9.7 Nonhomogeneous Linear Systems
   • 9.8 The Matrix Exponential Function
10. Partial Differential Equations
    • 10.1 Introduction: A Model for Heat Flow
    • 10.2 Method of Separation of Variables
    • 10.3 Fourier Series
    • 10.4 Fourier Cosine and Sine Series
    • 10.5 The Heat Equation
    • 10.6 The Wave Equation
    • 10.7 Laplace’s Equation

Appendix A Newton’s Method

Appendix B Simpson’s Rule

Appendix C Cramer’s Rule

Appendix D Method of Least Squares

Appendix E Runge-Kutta Procedure for n Equations