Ottawa: Department of Mathematics and Statistics University of Ottawa, 2011. — 317 pages. .
Notes for the course MAT 2384 3X.
Differential Equations and Laplace Transforms.
First-Order Ordinary Differential Equations.
Fundamental Concepts.
Separable Equations.
Equations with Homogeneous Coefficients.
Exact Equations.
Integrating Factors.
First-Order Linear Equations.
Orthogonal Families of Curves.
Direction Fields and Approximate Solutions.
Existence and Uniqueness of Solutions.
Second-Order Ordinary Differential Equations.
Linear Homogeneous Equations.
Homogeneous Equations with Constant Coefficients.
Basis of the Solution Space.
Independent Solutions.
Modeling in Mechanics.
Euler–Cauchy Equations.
Linear Differential Equations of Arbitrary Order.
Homogeneous Equations.
Linear Homogeneous Equations.
Linear Nonhomogeneous Equations.
Method of Undetermined Coefficients.
Particular Solution by Variation of Parameters.
Forced Oscillations.
Systems of Differential Equations.
Introduction.
Existence and Uniqueness Theorem.
Fundamental Systems.
Homogeneous Linear Systems with Constant Coefficients.
Nonhomogeneous Linear Systems.
Laplace Transform.
Definition.
Transforms of Derivatives and Integrals.
Shifts in s and in t.
Dirac Delta Function.
Derivatives and Integrals of Transformed Functions.
Laguerre Differential Equation.
Convolution.
Partial Fractions.
Transform of Periodic Functions.
Power Series Solutions.
The Method.
Foundation of the Power Series Method.
Legendre Equation and Legendre Polynomials.
Orthogonality Relations for Pn(x).
Fourier–Legendre Series.
Derivation of Gaussian Quadratures.
Numerical Methods.
Solutions of Nonlinear Equations.
Computer Arithmetic.
Review of Calculus.
The Bisection Method.
Fixed Point Iteration.
Newton’s, Secant, and False Position Methods.
Aitken–Steffensen Accelerated Convergence.
Horner’s Method and the Synthetic Division.
Muller’s Method.
Interpolation and Extrapolation.
Lagrange Interpolating Polynomial.
Newton’s Divided Difference Interpolating Polynomial.
Gregory–Newton Forward-Difference Polynomial.
Gregory–Newton Backward-Difference Polynomial.
Hermite Interpolating Polynomial.
Cubic Spline Interpolation.
Numerical Differentiation and Integration.
Numerical Differentiation.
The Effect of Roundoff and Truncation Errors.
Richardson’s Extrapolation.
Basic Numerical Integration Rules.
The Composite Midpoint Rule.
The Composite Trapezoidal Rule.
The Composite Simpson Rule.
Romberg Integration for the Trapezoidal Rule.
Adaptive Quadrature Methods.
Gaussian Quadrature.
Numerical Solution of Differential Equations.
Initial Value Problems.
Euler’s and Improved Euler’s Methods.
Low-Order Explicit Runge–Kutta Methods.
Convergence of Numerical Methods.
Absolutely Stable Numerical Methods.
Stability of Runge–Kutta Methods.
Embedded Pairs of Runge–Kutta Methods.
Multistep Predictor-Corrector Methods.
Stiff Systems of Differential Equations.
Exercises and Solutions.
Exercises for Differential Equations and Laplace Transforms.
Exercises for Chapter 1.
Exercises for Chapter 2.
Exercises for Chapter 3.
Exercises for Chapter 4.
Exercises for Chapter 5.
Exercises for Chapter 6.
Exercises for Numerical Methods.
Exercises for Chapter 7.
Exercises for Chapter 8.
Exercises for Chapter 9.
Exercises for Chapter 10.
Solutions to Starred Exercises.
Solutions to Exercises from Chapters 1 to 6.
Solutions to Exercises from Chapter 7.
Solutions to Exercises for Chapter 8.
Solutions to Exercises for Chapter 10.
Formulas and Tables.
Formulas and Tables.
Integrating Factor of M(x, y) dx + N(x, y) dy = 0.
Solution of First-Order Linear Differential Equations.
Laguerre Polynomials on 0 ≤ x ∞.
Legendre Polynomials Pn(x) on [−1, 1].
Fourier–Legendre Series Expansion.
Table of Integrals.
Table of Laplace Transforms.
