William F. Trench, 2013. — 806 p.Contents.
Introduction.
Applications Leading to Differential Equations.
First Order Equations.
Direction Fields for First Order Equations.
First Order Equations.
Linear First Order Equations.
Separable Equations.
Existence and Uniqueness of Solutions of Nonlinear Equations.
Transformation of Nonlinear Equations into Separable Equations.
Exact Equations.
Integrating Factors.
Numerical Methods.
Euler’s Method.
The Improved Euler Method and Related Methods.
The Runge-Kutta Method.
Applications of First Order Equations.
Growth and Decay.
Cooling and Mixing.
Elementary Mechanics.
Autonomous Second Order Equations.
Applications to Curves.
Linear Second Order Equations.
Homogeneous Linear Equations.
Constant Coefficient Homogeneous Equations.
Nonhomgeneous Linear Equations.
The Method of Undetermined Coefficients I.
The Method of Undetermined Coefficients II.
Reduction of Order.
Variation of Parameters.
Applcations of Linear Second Order Equations.
Spring Problems I.
Spring Problems II.
The RLC Circuit.
Motion Under a Central Force.
Series Solutions of Linear Second Order Equations.
Review of Power Series.
Series Solutions Near an Ordinary Point I.
Series Solutions Near an Ordinary Point II.
Regular Singular Points Euler Equations.
The Method of Frobenius I.
The Method of Frobenius II.
The Method of Frobenius III.
Laplace Transforms.
Introduction to the Laplace Transform.
The Inverse Laplace Transform.
Solution of Initial Value Problems.
The Unit Step Function.
Constant Coefficient Equations with Piecewise Continuous Forcing Functions.
Convolution.
Constant Cofficient Equations with Impulses.
A Brief Table of Laplace Transforms.
Linear Higher Order Equations.
Introduction to Linear Higher Order Equations.
Higher Order Constant Coefficient Homogeneous Equations.
Undetermined Coefficients for Higher Order Equations.
Variation of Parameters for Higher Order Equations.
Linear Systems of Differential Equations.
Introduction to Systems of Differential Equations.
Linear Systems of Differential Equations.
Basic Theory of Homogeneous Linear Systems.
Constant Coefficient Homogeneous Systems I.
Constant Coefficient Homogeneous Systems II.
Constant Coefficient Homogeneous Systems II.
Variation of Parameters for Nonhomogeneous Linear Systems.
Boundary Value Problems and Fourier Expansions.
Eigenvalue Problems for y''+lambda y = 0.
Fourier Series I.
Fourier Series II.
Fourier Solutions of Partial Differential Equations.
The Heat Equation.
The Wave Equation.
Laplace’s Equation in Rectangular Coordinates.
Laplace’s Equation in Polar Coordinates.
Boundary Value Problems for Second Order Linear Equations.
Boundary Value Problems.
Sturm–Liouville Problems.
