Written in terms of y we have y10 c0 y20 c1 ym0

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Unformatted text preview: 0) = c0 y2(0) = c1 ::: ym(0) = cm 1 : ; With DAEs and sometimes with IVPs and BVPs, it will be necessary to consider implicit di erential systems F(t y y ) = 0: (1.1.3) 0 If the Jacobian @ F=@ y is nonsingular then, by the implicit function theorem, (1.1.3) is equivalent to (1.1.1) however, it may be natural to solve (1.1.3) in its implicit form to maintain, e.g., sparsity. DAEs involve systems where @ F=@ y is singular. 0 0 7 Example 1.1.2. Let us write the oscillating pendelum problem as a rst-order system by letting y1 = x y2 = x y3 = y y4 = y : 0 0 Then, we have y1 = y2 0 T y2 = ; ml y1 T y4 = g ; ml y3 y3 = y4 0 0 with the constraint 0 2 2 y1 + y3 = l2 : This can be written in the form (1.1.3) with 2 y1 ; y2 1y 6 y2 + yml5 6 F(t y y ) = 6 y3 ; y4y y 6 3 4 y4 ; g + ml5 2 2 y1 + y3 ; l2 0 0 0 0 0 The Jacobian is clearly singular. y5 = T 2 100 60 1 0 6 Fy = 6 0 0 1 6 40 0 0 000 0 0 0 1 0 0 0 0 0 0 3 7 7 7 = 0: 7 5 3 7 7 7 7 5 Problems 1. Letting y1 = p and y2 = P , write the predator-prey model dp = p(a ; P...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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