The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
vi CONTENTS Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
Chapter 1 Classifying The Problem 1.1 Defining ODEs Definition 1.1 (Ordinary Differential Equation, ODE). Let y ∈ R n be a variable 1 that depends on t. Then we define a differential equation as any equation of the form F (t, y, y ′ ) = 0 (1.1) where F is any function of the 2n + 1 variables t, y 1 , y 2 , . . . , y n , y ′ 1 , y′ 2 , . . . , y′ n+1 . Definition 1.2. We will call any function φ(t) that satisfies a solution of the differential equation. F ( t, φ(t), φ ′ (t) ) = 0 (1.2) We will use the terms “Ordinary Differential Equation” and “Differential Equation”, as well as the abbreviations ODE and DE, interchangeably. More generally, on can include partial derivatives in the definition, in which case one must distinguish between “Partial” DEs (PDEs) and “Ordinary” DEs (ODEs). We will leave the study of PDEs to another class. Equation 1.1 is, in general, very difficult, and often, impossible, to solve, either analytically (e.g., by finding a formula that describes y), or numerically (e.g., for example, by using a computer to draw a picture of the graph of the solution). Often it is possible to solve 1.1 explicitly for the derivatives: y ′ = f(t, y) (1.3) Many important problems can be put into this form, and solutions are known to exists for a wide class of functions, particular as a result of theorem 1.3. The class of problems in which equation 1.1 can be converted to the form 1.3, at least locally, is not seriously restrictive from a practical point of view. The only requirements are 1 It may be either a scalar variable, in which case n = 1, or a vector variables, in which case n is the number of components of the vector. 1
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vi<br />
CONTENTS<br />
Math 582B, Spring 2007<br />
California State University Northridge<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007
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