This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Existence and Uniqueness Theorems for FirstOrder ODEs The general firstorder ODE is y = F ( x, y ) , y ( x ) = y . (*) We are interested in the following questions: (i) Under what conditions can we be sure that a solution to (*) exists? (ii) Under what conditions can we be sure that there is a unique solution to (*)? Here are the answers. Theorem 1 (Existence). Suppose that F ( x, y ) is a continuous function defined in some region R = { ( x, y ) : x < x < x + , y < y < y + } containing the point ( x , y ) . Then there exists a number 1 (possibly smaller than ) so that a solution y = f ( x ) to (*) is defined for x 1 < x < x + 1 . Theorem 2 (Uniqueness). Suppose that both F ( x, y ) and F y ( x, y ) are continuous functions defined on a re gion R as in Theorem 1. Then there exists a number 2 (possibly smaller than 1 ) so that the solution y = f ( x ) to (*), whose existence was guaranteed by Theorem 1, is the unique solution to (*) for x 2 < x < x + 2 ....
View Full
Document
 Spring '08
 LALALA
 Math

Click to edit the document details