Scientific Computing

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CS205 Homework #8 Solutions Problem 1 Give a criterion for the well-posedness of the k th order, scalar, homogeneous, constant-coefficient ODE u ( k ) + c k - 1 u ( k - 1) + · · · + c 1 u + c 0 u = 0 (Hint: Transform to a first-order system y = Ay and observe A is a matrix we’ve encountered previously in homework 3 problem 2) Solution Transforming the differential equation into a system of first order equations yields: u 1 u 2 . . . u k - 1 u k = u 2 u 3 . . . u k - k i =1 c i - 1 u i 1 = 0 1 0 · · · 0 0 0 0 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 1 0 - c 0 - c 1 - c 2 · · · - c k - 2 - c k - 1 u 1 u 2 . . . u k - 1 u k The matrix is a companion matrix as we saw in homework 3. Recall its characteristic polynomial is p ( λ ) = c 0 + c 1 λ + · · · + c k - 1 λ k - 1 + λ k . The eigenvalues of the matrix will be the roots of this polynomial. Thus, if the real parts of the roots are less than zero it is well-posed. If they are all not strictly less than zero then it is asymptotically stable. If any real part is positive then it is ill-posed.
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