existence and uniqueness

# existence and uniqueness - Existence and Uniqueness...

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Unformatted text preview: Existence and Uniqueness Theorems for First-Order ODE’s The general first-order 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- δ < x < x + δ, y- ² < y < y + ² } containing the point ( x , y ) . Then there exists a number δ 1 (possibly smaller than δ ) so that a solution y = f ( x ) to (*) is defined for x- δ 1 < x < x + δ 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 δ 2 (possibly smaller than δ 1 ) so that the solution y = f ( x ) to (*), whose existence was guaranteed by Theorem 1, is the unique solution to (*) for x- δ 2 < x < x + δ 2 ....
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existence and uniqueness - Existence and Uniqueness...

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