Schaums 題解精萃微分方程（影印版）

Chapter 1
BASIC CONCEPTS
Differential equations. Notation. Solutions. Initial-value and boundary-value
problems.
Chapter 2
CLASSIFICATIONS OF FIRST-ORDER DIFFERENTIAL
EQUATIONS
Standard form and differential form. Linear equations. Bernoulli equations.
Homogeneous equations. Separable equations. Exact equations.
Chapter 3
SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS
General solution. Solutions to the initial-value problem. Reduction of
homogeneous equations.
Chapter 4
EXACT FIRST-ORDER DIFFERENTIAL EQUATIONS
Defining properties. Method of solution. Integrating factors,
Chapter 5
LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS
Method of solution. Reduction of Bernoulli equations.
Chapter 6
APPLICATIONS OF FIRST-ORDER DIFFERENTIAL
EQUATIONS
Growth and decay problems. Temperature problems. Falling body problems.
Dilution problems. Electrical circuits. Orthogonal trajectories.
Chapter 7
LINEAR DIFFERENTIAL EQUATIONS: THEORY OF
SOLUTIONS
Linear differential equations. Linearly independent solutions. The Wronskian.
Nonhomogeneous equations.
Chapter 8
SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL
EQUATIONS WITH CONSTANT COEFFICIENTS
The characteristic equation. The general solution.
Chapter 9
nTH-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL
EQUATIONS WITH CONSTANT COEFFICIENTS
The characteristic equation. The general solution.
Chapter 10 THE METHOD OF UNDETERMINED COEFFICIENTS
Simple form of the method. Generalizations. Modifications. Limitations of
the method.
Chapter 11
VARIATION OF PARAMETERS
The method. Scope of the method.
Chapter 12
INITIAL-VALUE PROBLEMS
Chapter 13
APPLICATIONS OF SECOND-ORDER LINEAR
DIFFERENTIAL EQUATIONS
Spring problems. Electrical circuit problems. Buoyancy problems.
Classifying solutions.
Chapter 14
THE LAPLACE TRANSFORM
Definition. Properties of Laplace transforms. Functions of other
independent variables.
Chapter 15
INVERSE LAPLACE TRANSFORMS
Definition. Manipulating denominators. Manipulating numerators.
Chapter 16
CONVOLUTIONS AND THE UNIT STEP FUNCTION
Convolutions. Unit step function. Translations.
Chapter 17
SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS BY LAPLACE TRANSFORMS
Laplace transforms of derivatives. Solutions of differential equations.
Chapter 18
SOLUTIONS OF LINEAR SYSTEMS BY LAPLACE
TRANSFORMS
The method.
Chapter 19
MATRICES
Matrices and vectors. Matrix addition. Scalar and matrix multiplication. Powers
of a square matrix. Differentiation and integration of matrices. The
characteristic equation.
Chapter 20
eAt
Definition. Computation of eAt
Chapter 21
REDUCTION OF LINEAR DIFFERENTIAL EQUATIONS TO A
FIRST-ORDER SYSTEM
Reduction of one equation. Reduction of a system.
Chapter 22
SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH
CONSTANT COEFFICIENTS BY MATRIX METHODS
Solution of the initial-value problem. Solution with no initial conditions.
Chapter 23
LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE
COEFFICIENTS
Second-order equations. Analytic functions and ordinary points. Solutions
around the origin of homogeneous equations. Solutions around the origin of
nonbomogeneous equations. Initial-value problems. Solutions around other
points.
Chapter 24
REGULAR SINGULAR POINTS AND THE METHOD OF
FROBENIUS
Regular singular points. Method of Frobenius. General solution.
Chapter 25
GAMMA AND BESSEL FUNCTIONS
Gamma function. Bessel functions. Algebraic operations on infinite series.
Chapter 26
GRAPHICAL METHODS FOR SOLVING FIRST-ORDER
DIFFERENTIAL EQUATIONS
Direction fields. Euler''s method. Stability.
Chapter 27
NUMERICAL METHODS FOR SOLVING FIRST-ORDER
DIFFERENTIAL EQUATIONS
General remarks. Modified Euler''s method. Runge-Kutta method. Adams-
Bashforth-Moulton method. Milne''s method. Starting values. Order of a
numerical method.
Chapter 28
NUMERICAL METHODS FOR SYSTEMS
First-order systems. Euler''s method. Runge-Kutta method. Adams-Bashforth-
Moulton method.
Chapter 29
SECOND-ORDER BOUNDARY-VALUE PROBLEMS
Standard form. Solutions. Eigenvalue problems. Sturm-Liouville
problems. Properties of Sturm-Liouville problems.
Chapter 30
EIGENFUNCTION EXPANSIONS
Piecewise smooth functions. Fourier sine series. Fourier cosine series.
Appendix A LAPLACE TRANSFORMS
ANSWERS TO SUPPLEMENTARY PROBLEMS
INDEX