### Table of contents:

Part I: ORDINARY DIFFERENTIAL EQUATIONS

1. First-Order Differential Equations

Terminology and Separable Equations. Linear Equations. Exact Equations. Additional Applications. Existence and Uniqueness Questions. Direction Fields. Numerical Approximation of Solutions.

2. Linear Second-Order Equations

Theory of the Linear Second-Order Equation. The Constant Coefficient Homogeneous Equation. Solutions of the Nonhomogeneous Equation. Spring Motion.

3. The Laplace Transform

Definition and Notation. Solution of Initial Value Problems. Shifting and the Heaviside Function. Convolution. Impulses and the Dirac Delta Function. Appendix on Partial Fractions Decompositions.

4. Series Solutions

Power Series Solutions. Frobenius Solutions.

Part II: VECTORS, LINEAR ALGEBRA, AND SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS.

5. Algebra and Geometry of Vectors

Vectors in the Plane and 3-Space. The Dot Product. The Cross Product. The Vector Space Rn.

6. Matrices and Systems of Linear Equations

Matrices. Linear Homogeneous Systems. Nonhomogeneous Systems of Linear Equations. Matrix Inverses.

7. Determinants

Definition of the Determinant. Evaluation of Determinants I. Evaluation of Determinants II. A Determinant Formula for A−1. Cramer’s Rule.

8. Eigenvalues and Diagonalization

Eigenvalues and Eigenvectors. Diagonalization. Some Special Matrices.

9. Systems of Linear Differential Equations

Systems of Linear Differential Equations. Solution of X_ = AX when A Is Constant. Solution of X_ = AX + G.

Part III: VECTOR ANALYSIS

10. Vector Differential Calculus

Vector Functions of One Variable. Velocity and Curvature. Vector Fields and Streamlines. The Gradient Field. Divergence and Curl.

11. Vector Integral Calculus

Line Integrals. Green’s Theorem. An Extension of Green’s Theorem. Potential Theory. Surface Integrals. Applications of Surface Integrals. The Divergence Theorem of Gauss. Stokes’s Theorem.

Part IV: FOURIER ANALYSIS AND EIGENFUNCTION EXPANSIONS

12. Fourier Series

The Fourier Series of a Function. Sine and Cosine Series. Derivatives and Integrals of Fourier Series. Complex Fourier Series.

13. The Fourier Integral and Transforms

The Fourier Integral. Fourier Cosine and Sine Integrals. The Fourier Transform. Fourier Cosine and Sine Transforms.

14 Eigenfunction Expansions

General Eigenfunction Expansions. Fourier-Legendre Expansions. Fourier-Bessel Expansions.

Part V: PARTIAL DIFFERENTIAL EQUATIONS

15. The Wave Equation

Derivation of the Equation. Wave Motion on an Interval. Wave Motion in an Infinite Medium. Wave Motion in a Semi-Infinite Medium. d’Alembert’s Solution. Vibrations in a Circular Membrane. Vibrations in a Rectangular Membrane.

16. The Heat Equation

Initial and Boundary Conditions. The Heat Equation on [0, L]. Solutions in an Infinite Medium. Heat Conduction in an Infinite Cylinder. Heat Conduction in a Rectangular Plate.

17. The Potential Equation

Laplace’s Equation. Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. Poisson’s Integral Formula. Dirichlet Problem for Unbounded Regions. A Dirichlet Problem for a Cube. Steady-State Heat Equation for a Sphere.

APPENDIX A Guide to Notation

APPENDIX B A MAPLE Primer

Answers to Selected Problems

Index

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