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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 tx-plane, 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 tx-plane, ( 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...
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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