Boyce/DiPrima 9 th ed, Ch1.3: Classification of Differential Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.

Slides:



Advertisements
Similar presentations
Boyce/DiPrima 9th ed, Ch 2.4: Differences Between Linear and Nonlinear Equations Elementary Differential Equations and Boundary Value Problems, 9th edition,
Advertisements

Boyce/DiPrima 9th ed, Ch 2.8: The Existence and Uniqueness Theorem Elementary Differential Equations and Boundary Value Problems, 9th edition, by William.
Boyce/DiPrima 9th ed, Ch 2.7: Numerical Approximations: Euler’s Method Elementary Differential Equations and Boundary Value Problems, 9th edition, by.
Boyce/DiPrima 9th ed, Ch 3.5: Nonhomogeneous Equations;Method of Undetermined Coefficients Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 10th ed, Ch 10.1: Two-Point Boundary Value Problems Elementary Differential Equations and Boundary Value Problems, 10th edition, by William.
Boyce/DiPrima 9th ed, Ch 2.5: Autonomous Equations and Population Dynamics Elementary Differential Equations and Boundary Value Problems, 9th edition,
Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Seven Ordinary Differential Equations.
Ordinary Differential Equations S.-Y. Leu Sept. 21, 2005.
Ordinary Differential Equations S.-Y. Leu Sept. 21,28, 2005.
1 Chapter 9 Differential Equations: Classical Methods A differential equation (DE) may be defined as an equation involving one or more derivatives of an.
Differential Equations
CISE301_Topic8L1KFUPM1 CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1.
Introduction 1. MSc (1994): Punjab University Specialization: Applied Mathematics 2. MS /M.Phil (2006): COMSATS, Islamabad Specialization: Applied Mathematics.
Chapter 1: First-Order Differential Equations 1. Sec 1.1: Differential Equations and Mathematical Models Definition: Differential Equation An equation.
Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Elementary Differential Equations and Boundary Value Problems, 9 th edition,
Boyce/DiPrima 9 th ed, Ch 3.1: 2 nd Order Linear Homogeneous Equations-Constant Coefficients Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 10th ed, Ch 10.5: Separation of Variables; Heat Conduction in a Rod Elementary Differential Equations and Boundary Value Problems, 10th.
Boyce/DiPrima 9th ed, Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 9th ed, Ch 3.4: Repeated Roots; Reduction of Order Elementary Differential Equations and Boundary Value Problems, 9th edition, by William.
Boyce/DiPrima 9 th ed, Ch 2.7: Numerical Approximations: Euler’s Method Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.
Boyce/DiPrima 9th ed, Ch 8.4: Multistep Methods Elementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce and Richard.
Boyce/DiPrima 9 th ed, Ch 11.5: Further Remarks on Separation of Variables: A Bessel Series Expansion Elementary Differential Equations and Boundary Value.
Fin500J Topic 6Fall 2010 Olin Business School 1 Fin500J: Mathematical Foundations in Finance Topic 6: Ordinary Differential Equations Philip H. Dybvig.
Ordinary Differential Equations
Math 3120 Differential Equations with Boundary Value Problems
Boyce/DiPrima 9 th ed, Ch 5.1: Review of Power Series Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce.
Boyce/DiPrima 9th ed, Ch 4.2: Homogeneous Equations with Constant Coefficients Elementary Differential Equations and Boundary Value Problems, 9th edition,
Boyce/DiPrima 9 th ed, Ch 2.7: Numerical Approximations: Euler’s Method Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.
Boyce/DiPrima 9 th ed, Ch 1.4: Historical Remarks Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and.
Boyce/DiPrima 9 th ed, Ch 10.8: Laplace’s Equation Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and.
Boyce/DiPrima 9 th ed, Ch 3.2: Fundamental Solutions of Linear Homogeneous Equations Elementary Differential Equations and Boundary Value Problems, 9 th.
Boyce/DiPrima 9th ed, Ch 1.2: Solutions of Some Differential Equations Elementary Differential Equations and Boundary Value Problems, 9th edition, by.
Boyce/DiPrima 9th ed, Ch 4.1: Higher Order Linear ODEs: General Theory Elementary Differential Equations and Boundary Value Problems, 9th edition, by.
Ch 1.3: Classification of Differential Equations
Boyce/DiPrima 9 th ed, Ch 6.2: Solution of Initial Value Problems Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William.
Boyce/DiPrima 9th ed, Ch 3.3: Complex Roots of Characteristic Equation Elementary Differential Equations and Boundary Value Problems, 9th edition, by.
1 Chapter 1 Introduction to Differential Equations 1.1 Introduction The mathematical formulation problems in engineering and science usually leads to equations.
Differential Equations
Boyce/DiPrima 9 th ed, Ch 2.2: Separable Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and.
Ch 1.1: Basic Mathematical Models; Direction Fields Differential equations are equations containing derivatives. The following are examples of physical.
Boyce/DiPrima 9 th ed, Ch 11.3: Non- Homogeneous Boundary Value Problems Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.
Section 1.1 Basic Definitions and Terminology. DIFFERENTIAL EQUATIONS Definition: A differential equation (DE) is an equation containing the derivatives.
Boyce/DiPrima 9 th ed, Ch 6.1: Definition of Laplace Transform Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William.
Boyce/DiPrima 9 th ed, Ch 5.3: Series Solutions Near an Ordinary Point, Part II Elementary Differential Equations and Boundary Value Problems, 9 th edition,
Boyce/DiPrima 9 th ed, Ch 6.3: Step Functions Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and Richard.
Boyce/DiPrima 9 th ed, Ch 2.6: Exact Equations & Integrating Factors Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William.
Ordinary Differential Equations (ODEs). Objectives of Topic  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods.
Introduction to Differential Equations
Boyce/DiPrima 10th ed, Ch 6.2: Solution of Initial Value Problems Elementary Differential Equations and Boundary Value Problems, 10th edition, by William.
SLOPE FIELDS & EULER’S METHOD
Boyce/DiPrima 10th ed, Ch 10.3: The Fourier Convergence Theorem Elementary Differential Equations and Boundary Value Problems, 10th edition, by William.
Basic Definitions and Terminology
Boyce/DiPrima 10th ed, Ch 6.3: Step Functions Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce and.
Boyce/DiPrima 9th ed, Ch 2.7: Numerical Approximations: Euler’s Method Elementary Differential Equations and Boundary Value Problems, 9th edition, by.
Boyce/DiPrima 10th ed, Ch 7.4: Basic Theory of Systems of First Order Linear Equations Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 10th ed, Ch 6.1: Definition of Laplace Transform Elementary Differential Equations and Boundary Value Problems, 10th edition, by William.
525602:Advanced Numerical Methods for ME
Ch 1.3: Classification of Differential Equations
Ch 1.3: Classification of Differential Equations
Sec 5.5:Variation of Parameters
Boyce/DiPrima 10th ed, Ch 6.4: Differential Equations with Discontinuous Forcing Functions Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 9th ed, Ch 4.3: Nonhomogeneous Equations: Method of Undetermined Coefficients Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 10th ed, Ch 7.1: Introduction to Systems of First Order Linear Equations Elementary Differential Equations and Boundary Value Problems,
Boyce/DiPrima 9th ed, Ch 3.6: Variation of Parameters Elementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce.
Introduction to Ordinary Differential Equations
Chapter 1: Introduction to Differential Equations
CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1 CISE301_Topic8L1 KFUPM.
Chapter 1: First-Order Differential Equations
Chapter 1:First order Partial Differential Equations
Presentation transcript:

Boyce/DiPrima 9 th ed, Ch1.3: Classification of Differential Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc. The main purpose of this course is to discuss properties of solutions of differential equations, and to present methods of finding solutions or approximating them. To provide a framework for this discussion, in this section we give several ways of classifying differential equations.

Ordinary Differential Equations When the unknown function depends on a single independent variable, only ordinary derivatives appear in the equation. In this case the equation is said to be an ordinary differential equations (ODE). The equations discussed in the preceding two sections are ordinary differential equations. For example,

Partial Differential Equations When the unknown function depends on several independent variables, partial derivatives appear in the equation. In this case the equation is said to be a partial differential equation (PDE). Examples:

Systems of Differential Equations Another classification of differential equations depends on the number of unknown functions that are involved. If there is a single unknown function to be found, then one equation is sufficient. If there are two or more unknown functions, then a system of equations is required. For example, predator-prey equations have the form where u(t) and v(t) are the respective populations of prey and predator species. The constants a, c, ,  depend on the particular species being studied. Systems of equations are discussed in Chapter 7.

Order of Differential Equations The order of a differential equation is the order of the highest derivative that appears in the equation. Examples: We will be studying differential equations for which the highest derivative can be isolated:

Linear & Nonlinear Differential Equations An ordinary differential equation is linear if F is linear in the variables Thus the general linear ODE has the form Example: Determine whether the equations below are linear or nonlinear.

Solutions to Differential Equations A solution  (t) to an ordinary differential equation satisfies the equation: Example: Verify the following solutions of the ODE

Solutions to Differential Equations Three important questions in the study of differential equations: Is there a solution? (Existence) If there is a solution, is it unique? (Uniqueness) If there is a solution, how do we find it? (Analytical Solution, Numerical Approximation, etc)