Linear Algebra and Ordinary Differential Equations – Alan Jeffrey – 1st Edition

Pan 1: Mathematical Prerequisites
1. Review of Topics from Analysis
1.2 Intervals and inequalities
1.3 Mathematical induction
1.4 Polynomials and partial fractions
1.6 Complex numbers in Cartesian form
1.7 Complex numbers in polar form. Roots
1.8 Some properties of integrals
1.9 Linear difference equations

Part 2: Vectors and Linear Algebra
2. Algebra of Vectors
2.3 Vectors - a geometrical approach in R
2.4 Vectors in component form
2.5 Scalar product (dot product)
2.6 Vector product (cross product)
2.7 Combinations of scalar and vector products
2.8 Geometrical applications of scalar and vector products
2.9 Vector spaces
3. Matrices
3.2 Addition of matrices, multiplication by a number and the transposition operation
3.3 Matrix multiplication. Linear transformations. Differentiation
3.4 Systems of linear equations. Solution by elimination
3.5 Linear independence. Rank. Reduced echelon form
3.7 Determinants
3.8 Determinants and rank. Cramer's rule
3.9 Inverse matrices
3.10 Algebraic eigenvalue problems. Eigenvalues
3.11 Diagonalizability of matrices. The Cayley—Hamilton theorem
3.12 Quadratic forms
3.13 The LU and Cholesky factorization methods

Part 3: Ordinary Differential Equations
4. First Order Ordinary Differential Equations
4.1 Differential equations and their origins
4.2 First order differential equations and isoclines
4.3 Separable equations
4.4 Exact differential equations and integrating factors
4.5 Linear first order differential equations
4.6 Orthogonal and isogonal trajectories
4.7 Existence, uniqueness and an iterative method of solution
4.8 Numerical solution of first order equations by the Runge-Kutta method
5. Linear Higher Order Ordinary Differential Equations
5.1 Linear higher order ordinary differential equations
5.2 Second order constant coefficient equations — homogeneous case
5.3 Higher order constant coefficient equations — homogeneous case
5.4 Differential operators
5.5 Nonhomogeneous linear differential equations
5.6 General reduction of the order of a linear differential equation. Integral method
5.7 Oscillatory behavior
5.8 Reduction to the normal form u"-f(x)u=0
5.9 The Green's function
6. Systems of Linear Differential Equations
6.1 First order linear homogeneous systems of differential equations
6.2 First order linear nonhomogeneous systems of differential equations
6.3 Second order linear systems of differential equations
6.4 Qualitative theory: the phase plane and stability
6.5 Numerical solution of systems by the Runge-Kutta method
7. Laplace Transform and z-transform
7.1 The Laplace transform — introductory ideas
7.2 Operational properties of the Laplace transform
7.3 Applications of the Laplace transform
7.4 The z-transform
7.5 Applications of the z—transform
8. Series Solution of Ordinary Differential Equations
8.1 Sequences, convergence and power series
8.2 Solving differential equations by Taylor series
8.3 Solution in the neighborhood of an ordinary point
8.4 Legendre's equation and Legendre polynomials
8.5 The gamma function T(x)
8.6 Frobenius' method and its extension
8.7 Bessel functions
8.8 Asymptotic expansions
8.9 Numerical solution of second order equations by the Runge—Kutta method
9. Fourier Series, Sturm-Liouville Problems and Orthogonal Functions
9.1 Trigonometric series, periodic extension and convergence
9.2 The formal development of Fourier series
9.3 Convergence of Fourier series and related results
9.4 Integration and differentiation of Fourier series
9.7 Numerical harmonic analysis
9.8 Representation of functions using orthogonal systems. Sturm-Liouville problems
9.9 Expansions in terms of Bessel functions
9.10 Orthogonal polynomials