PART I: ORDINARY DIFFERENTIAL EQUATIONS. 1. First-Order Differential Equations. Terminology and Separable Equations. Singular Solutions, Linear Equations. Exact Equations. Homogeneous, Bernoulli and Riccati Equations. 2. Second-Order Differential Equations. The Linear Second-Order Equation. The Constant Coefficient Homogeneous Equation. Particular Solutions of the Nonhomogeneous Equation. The Euler Differential Equation, Series Solutions. Frobenius Series Solutions. 3. The Laplace Transform. Definition and Notation. Solution of Initial Value Problems. The Heaviside Function and Shifting Theorems. Convolution. Impulses and the Dirac Delta Function. Systems of Linear Differential Equations. 4. Eigenfunction Expansions. Eigenvalues, Eigenfunctions, and Sturm-Liouville Problems. Eigenfunction Expansions, Fourier Series. Part II: PARTIAL DIFFERENTIAL EQUATIONS. 5. The Heat Equation. Diffusion Problems in a Bounded Medium. The Heat Equation with a Forcing Term F(x,t). The Heat Equation on the Real Line. A Reformulation of the Solution on the Real Line. The Heat Equation on a Half-Line, The Two-Dimensional Heat Equation. 6. The Wave Equation. Wave Motion on a Bounded Interval. The Effect of c on the Motion. Wave Motion with a Forcing Term F(x). Wave Motion in an Unbounded Medium. The Wave Equation on the Real Line. dAlemberts Solution and Characteristics. The Wave Equation with a Forcing Term K(x,t). The Wave Equation in Higher Dimensions. 7. Laplaces Equation. The Dirichlet Problem for a Rectangle. Dirichlet Problem for a Disk. The Poisson Integral Formula. The Dirichlet Problem for Unbounded Regions. A Dirichlet Problem in 3 Dimensions. The Neumann Problem. Poissons Equation. 8. Special Functions and Applications. Legendre Polynomials. Bessel Functions. Some Applications of Bessel Functions. 9. Transform Methods of Solution. Laplace Transform Methods. Fourier Transform Methods. Fourier Sine and Cosine Transforms. Part III: MATRICES AND LINEAR ALGEBRA. 10. Vectors and the Vector Space Rn. Vectors in the Plane and 3 Space. The Dot Product. The Cross Product. n-Vectors and the Algebraic Structure of Rn. Orthogonal Sets and Orthogonalization. Orthogonal Complements and Projections. 11. Matrices, Determinants and Linear Systems. Matrices and Matrix Algebra. Row Operations and Reduced Matrices. Solution of Homogeneous Linear Systems. Solution of Nonhomogeneous Linear Systems. Matrix Inverses. Determinants, Cramers Rule. The Matrix Tree Theorem. 12. Eigenvalues, Diagonalization and Special Matrices. Eigenvalues and Eigenvectors. Diagonalization. Special Matrices and Their Eigenvalues and Eigenvectors. Quadratic Forms. PART IV: SYSTEMS OF DIFFERENTIAL EQUATIONS. 13. Systems of Linear Differential Equations. Linear Systems. Solution of X = AX When A Is Constant. Exponential Matrix Solutions. Solution of X = AX + G for Constant A. 14. Nonlinear Systems and Qualitative Analysis. Nonlinear Systems and Phase Portraits. Critical Points and Stability. Almost Linear Systems, Linearization. Part V: VECTOR ANALYSIS. 15. Vector Differential Calculus. Vector Functions of One Variable. Velocity, Acceleration, and Curvature. The Gradient Field. Divergence and Curl. Streamlines of a Vector Field. 16. Vector Integral Calculus. Line Integrals. Greens Theorem. Independence of Path and Potential Theory. Surface Integrals. Applications of Surface Integrals. Gausss Divergence Theorem. Stokess Theorem. PART VI: FOURIER ANALYSIS. 17. Fourier Series. Fourier Series On [-L, L]. Fourier Sine and Cosine Series. Integration and Differentiation of Fourier Series. Properties of Fourier Coefficients. Phase Angle Form. Complex Fourier Series, Filtering of Signals. 18. Fourier Transforms. The Fourier Transform. Fourier Sine and Cosine Transforms. PART VII: COMPLEX FUNCTIONS. 19. Complex Numbers and Functions. Geometry and Arithmetic of Complex Numbers. Complex Functions, Limits. The Exponential and Trigonometric Functions. The Complex Logarithm. Powers. 20. Integration. The Integral of a Complex Function. Cauchys Theorem. Consequences of Cauchys Theorem. 21. Series Representations of Functions. Power Series. The Laurent Expansion. 22. Singularities and the Residue Theorem. Classification of Singularities. The Residue Theorem. Evaluation of Real Integrals. 23. Conformal Mappings. The Idea of a Conformal Mapping. Construction of Conformal Mappings. Notation. ANSWERS TO SELECTED PROBLEMS.
