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# To ensure that this is so we must supplement the odes

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Unformatted text preview: o ensure that this is so, we must supplement the ODEs by the algebraic constraint x2 + y2 = l2 : The two second-order di erential equations and the constraint comprise a system of ve di erential algebraic equations (DAEs) for the unknowns x(t), y(t), and T (t). Initial conditions specify x(0), dx(0)=dt, y(0), and dy(0)=dt, but not T (0). 5 Of course, for this problem, it's easy to eliminate the constraint by introducing the change of variables x = l sin , y = l cos . This would reduce the DAEs to the second-order ODE d2 = ; g sin : dt2 l Such simpli cations would not be possible with more di cult systems. θ y T x mg Figure 1.1.3: Oscillations of a simple Pendelum. IVPs will comprise our initial study (Part 2 of these notes). We'll take up BVPs next (Part 3) and conclude with a study of DAEs (Part 4). In almost all cases, it will su ce to develop and analyze numerical methods for IVPs and BVPs that are written as rst-order vector systems of ODEs in the explicit form y (t) = f (t y) 0 t>0 (1.1.1a...
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