Existence and Uniqueness Theorems for First-Order ODE’s

Existence and Uniqueness Theorems for First-Order ODE's The general First-Order ODE is For a real number x and a positive value , the set of numbers x satisfying x0 < x < x0 + is called an y 0 = F (x, y), y(x0 ) = y0 . (*) open interval centered at x0 . 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 Example 3. Consider the ODE. R = {(x, y) : x0 < x < x0 + , y0 < y < y0 + } y 0 = x y + 1, y(1) = 2. containing the point (x0 , y0 ). Then there exists a number In this case, both the function F (x, y) = x y +1 and its F.

Existence and Uniqueness Theorems for First-Order ODE’s The general rst-order ODE is y0 = F(x;y); y(x0) = y0: (*) We are interested in the following questions: (i) Under what conditions can we be sure that a solution

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  Theorem, Existence, Uniqueness, Existence and uniqueness theorems

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