Elementary Differential Equations by William Trench

1. INTRODUCTION Some Applications Leading to Differential Equations / Basic Concepts / Direction Fields for First Order Equations
2.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
3.NUMERICAL METHODS Euler's Method / The Improved Euler Method and Related Methods / The Runge-Kutta Method
4.APPLICATIONS OF FIRST ORDER EQUATIONS Growth and Decay / Cooling and Mixing / Elementary Mechanics / Autonomous Second Order Equations / Applications to Curves
5.LINEAR SECOND ORDER EQUATIONS Homogeneous Linear Equations / Constant Coefficient Homogeneous Equations / Nonhomogenous Linear Equations / The Method of Undetermined Coefficients I / The Method of Undetermined Coefficients II / Reduction of Order / Variation of Parameters
6.APPLICATIONS OF LINEAR SECOND ORDER EQUATIONS Spring Problems I / Spring Problems II / The RLC Circuit / Motion Under a Central Force
7.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
8.LAPLACE TRANSFORMS Introduction to the Laplace Transform / The Inverse Laplace Transform / Solution of the Initial Value Problems / The Unit Step Function / Constant Coefficient Equations with Piecewise Continuous Forcing Functions / Convolution / Constant Coefficient Equations with Impulses
9.LINEAR HIGHER ORDER EQUATIONSIntroduction to Linear Higher Order Equations / Higher Order Constant Coefficient Homogeneous Equations / Undetermined Coefficients for Higher Order Equations / Variation of Parameters for Higher Order Equations
10.LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS Introduction to Systems of Differential Equations / Linear Systems of Differential Equations / Basic Theory of Homogenous Linear Systems / Constant Coefficient Homogeneous Systems I / Constant Coefficient Homogenous Systems II / Constant Coefficient Homogenous Systems III / Variation of Parameters for Nonhomogenous Linear Systems / ANSWERS TO SELECTED EXERCISES / INDEX