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Unformatted text preview: CS205 Homework #8 Review Session Notes Properties of First Order ODEs Most properties of first order scalar ODEs ( y = λy ) extend naturally to their vectorvalued counterparts ( y = Ay ). 1. Existence and Uniqueness . Consider the following scalar and vectorvalued initial value problems: y = λy, y ( t ) = y y = Ay , y ( t ) = y In both cases, a unique solution exists. 2. WellPosedness . We say that an initial value problem is strictly wellposed if its analytic solution decays to zero as t → ∞ , regardless of the initial value condition itself. For the scalar ODE y = λy the necessary and sufficient condition for well posedness is λ < 0. For the vectorvalued case, the corresponding condition is that Re { λ } < 0 for any eigenvalue λ of A . There is also a “relaxed” definition for wellposedness, which is much less common. The relaxed definition requires only that the analytic solution to an initial value problem remains bounded , regardless of the initial conditions. The necessary and sufficient, regardless of the initial conditions....
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 Fall '07
 Fedkiw
 Linear Algebra, Numerical Analysis, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix

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