Sec_2.1 - Ch2 First-Order Differential Equation 2.1...

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Ch2. First-Order Differential Equation 2.1 Preliminary Theory: Initial-Value Problem We are often interested in solving the first-order ODE , dy f x y dx subject to the condition 0 0 y x y . This type of problem is called IVP. Ex1. In chapter 1, we have seen that x y c e is a one-parameter family of solutions for y y   on the interval ,   . If we specify the condition to be 0 2 y , then 0 0 2 2 . x y c e c y x e It is a solution of the IVP. Ex2. The functions 0 y and 4 /16 y x satisfies the differential equation and initial condition in the problem 1 2 , 0 0 dy xy y dx . In this class, it is often to know whether a solution exists and, when it does, whether it is unique. Theorem 2.1 existence of a Unique Solution Let R be a rectangular region in the xy -plane defined by
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Unformatted text preview: a x b , c y d containing the point , x y in the interior. If , f x y and / f y are continuous on R , then there exist an interval I centered at x and a unique function y x defined on I satisfying the IVP. If the condition does not hold, the IVP may have no solution or more than one solution. Ex3. Using Theorem 2.1 The theorem guarantees that there exists an interval about x on which 2 x y e is the only solution of the initial value problem of , 2 y y y . It is clear that , , / 1 f x y y f y are continuous everywhere on the xy-plane so there exists a unique solution. Homework:1~15 odd....
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  • Fall '06
  • EDeSturler
  • Equations, Boundary value problem, Lipschitz continuity, first-order differential equation

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