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Section 2.1

Section 2.1. Preliminary Theory. INITIAL-VALUE PROBLEM. Most first-order DEs can be written in the form When asked to solve (i) given the side condition that y ( x 0 ) = y 0 (ii),

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Section 2.1

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  1. Section 2.1 Preliminary Theory

  2. INITIAL-VALUE PROBLEM Most first-order DEs can be written in the form When asked to solve (i) given the side condition that y(x0) = y0 (ii), we call the problem, (i) and (ii), an initial-value problem (IVP). The side condition is known as an initial condition.

  3. TWO QUESTIONS CONCERNING AN IVP • Does a solution to the problem exist? • If a solution exists, is it unique?

  4. EXISTENCE OF AUNIQUE SOLUTION Theorem: Let R be a rectangular region in the xy-plane defined by a≤ x≤ b, c ≤ y ≤ d that contains the point (x0, y0) in its interior. If f (x, y) and ∂f /∂y are continuous on R, then there exists an interval I centered at x0 and a unique function y(x) defined on I satisfying the initial-value problem.

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