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: LAB #3 The Existence and Uniqueness Theorems Goal : Determine under what circumstances solution curves for a differential equation x = f ( t, x ) can cross; examine when can a solution not exist and when there are multiple solutions; use Existence and Uniqueness Theorems to derive qualitative information about solutions. Required tools : dfield ; Existence and Uniqueness Theorems (Theorem 2.4.2 in § 2.4 and 2.8.1 in § 2.8 of the text); separable differential equations. Discussion The following constitute the Existence and Uniqueness Theorems from the text: Existence Theorem : If f ( t, x ) is defined and continuous on a rectangle R in the txplane, then given any point ( t , x ) ∈ R , the initial value problem x = f ( t, x ) and x ( t ) = x has a solution x ( t ) defined in an interval containing t . Uniqueness Theorem : If f ( t, x ) and ∂f ∂x are both continuous on a rectangle R in the txplane, ( t , x ) ∈ R , and if both x ( t ) and y ( t ) satisfy the same initial value problem x = f ( t, x ) and x ( t ) = x , then as long as ( t, x ( t )) and ( t, y ( t )) stay in R , we have x ( t ) = y ( t ) . (The Uniqueness Theorem asserts that if f ( t, x ) and ∂f ∂x are continuous on a rectangle R , then solutions to the differential equation x = f ( t, x ) cannot cross in R .) Assignment (1) Choose a differential equation...
View
Full
Document
This note was uploaded on 04/23/2011 for the course MA 366 taught by Professor Cho during the Spring '08 term at Purdue.
 Spring '08
 CHO

Click to edit the document details