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# Letting y1 p and y2 p write the predator prey model dp

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Unformatted text preview: ) dt dP = P (;c + p) dt in the vector form (1.1.1). t>0 p(0) = p0 t>0 P (0) = P0 2. Write the mechanical vibration problem 2 m d y + c dy + ky = f (t) dt2 dt t>0 in the form (1.1.1a-c) 8 y(0) = y0 dy(0) = y 0 dt 0 3. Write the column buckling problem d4y + d2y = 0 dt4 dt2 y(0) = dy(0) = 0 dt 0<t<l d2 y(l) = 0 y(l) = dt2 in the form (1.1.1a,b,d) 1.2 Existence, Uniqueness, and Stability of IVPs Before considering numerical procedures for (1.1.1, 1.1.3), let us review some results from ODE theory that ensure the existence of unique solutions which depend continuously on the initial data. For simplicity, let us focus on a scalar IVP having the form (1.1.2). De nition 1.2.1. A function f (t y) satis es a Lipschitz condition in a domain D if there exists a non-negative constant L such that jf (t y) ; f (t z)j Ljy ; zj 8(t y) (t z) 2 D: (1.2.4) Satisfying a Lipschitz condition guarantees that solutions y(t) of (1.1.2) are unique as expressed by the following theorem. Theorem 1.2.1. Let f (t y...
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