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


Appendices

-Newton’s Method
-Simpson’s Rule
-Cramer’s Rule
-Method of Least Squares
-Runge-Kutta Procedure for n Equations