Differential Equations With Boundary-value Problems - 4th edition

1. INTRODUCTION TO DIFFERENTIAL EQUATIONS.

Definitions and Terminology.
Initial-Value Problems.
Differential Equations as Mathematical Models.

2. FIRST-ORDER DIFFERENTIAL EQUATIONS.

Separable Variables.
Exact Equations.
Linear Equations.
Solutions by Substitutions.

3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS.

Linear Equations.
Nonlinear Equations.
Systems of Linear and Nonlinear Equations.

4. DIFFERENTIAL EQUATIONS OF HIGHER ORDER.

Preliminary Theory : Linear Equations.
Reduction of Order.
Homogeneous Linear Equations with Constant Coefficients.
Undetermined Coefficients - Superposition Approach.
Undetermined Coefficients - Annihilator Approach.
Variation of Parameters.
Cauchy-Euler Equations.
Systems of Linear Equations.
Nonlinear Equations.

5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS.

Linear Equations : Initial-Value Problems.
Linear Equations : Boundary-Value.
Problems.
Nonlinear Equations.

6. SERIES SOLUTIONS OF LINEAR EQUATIONS.

Review of Power Series.
Power Series Solutions.
Solutions about Ordinary Points.
Solutions about Singular Points.
Two Special Equations.

7. LAPLACE TRANSFORM.

Definition of the Laplace Transform.
Inverse Transform.
Translation Theorems and Derivatives of a Transform.
Transforms of Derivatives, Integrals and Periodic Functions.
Applications.
Dirac Delta Function.
Systems of Linear Equations.

8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.

Preliminary Theory.
Homogeneous Linear Systems with Constant Coefficients.
Variation of Parameters.
Matrix Exponential.

9. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS.

Direction Fields.
Euler Methods.
Runge-Kutta Methods.
Multistep Methods.
Higher-Order Equations and Systems.
Second-Order Boundary-Value Problems.

10. PLANE AUTONOMOUS SYSTEMS AND STABILITY.

Autonomous Systems, Critical Points, and Periodic Solutions.
Stability of Linear Systems.
Linearization and Local Stability.
Modeling Using Autonomous Systems.

11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES.

Orthogonal Functions.
Fourier Series.
Fourier Cosine and Sine Series.
Sturm-Liouville Problem.
Bessel and Legendre Series.

12. PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES.

Separable Partial Differential Equations.
Classical Equations and Boundary-Value Problems.
Heat Equation.
Wave Equation.
Laplace's Equation.
Nonhomogeneous Equations and Boundary Conditions.
Use of Generalized Fourier Series.
Boundary-Value Problems Involving Fourier Series in Two Variables.

13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS.

Problems involving Laplace's Equation in Polar Coordinates.
Problems in Polar and Cylindrical Coordinates : Bessel Functions.
Problems in Spherical Coordinates : Legendre Polynomials.

14. INTEGRAL TRANSFORM METHOD.

Error Function.
Applications of the Laplace Transform.
Fourier Integral.
Fourier Transforms.

15. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS.

Elliptic Equations.
Parabolic Equations.
Hyperbolic Equations.

Appendix I : Gamma Function.
Appendix II : Introduction to Matrices.
Appendix III : Laplace Transforms.

Answers to Odd-Numbered Problems.
Index.