Question: Prove the proposition: Let U R^d be open, and let F : U R^d be continuous. If x C^1((, ), U) satisfies the IVP such

Prove the proposition:

Let U R^d be open, and let F : U R^d be continuous. If x C^1((, ), U) satisfies the IVP such that x'=F(x), x(0)=b, then x satisfies the integral relation x(t) = b + integral (F(x(s)) ds, from 0 to t, < t < . Conversely, if x is continuous on (, ) and satisfies the integral, then x is C^1 and satisfies the IVP.

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