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The Method of Successive Approximations. Consider the in

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The Method of Successive Approximations. Consider the initial value problem x? = Ax, x(0) = x0, (i) where A is a constan

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The Method of Successive Approximations. Consider the initial value problem x? = Ax, x(0) = x0, (i) where A is a constant matrix and x0 is a prescribed vector. (a) Assuming that a solution x = ?(t) exists, show that it must satisfy the integral equation ?(t) = x0 + ? t 0 A?(s) ds. (ii) (b) Start with the initial approximation ?(0) = x0. Substitute this expression for ?(t) in the right-hand side of Eq. (ii) and obtain a new approximation ?(1)(t). Show that ?(1)(t) = (In + At)x0. (iii) (c) Repeat the process and thereby obtain a sequence of approximations ?(0),?(1),?(2),

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