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Consider an oscillator satisfying the initial value prob

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Consider an oscillator satisfying the initial value problem u?? + ?2 u = 0, u(0) = u0, u? (0) = ?0. (i) (a) Let x1 = u,

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Consider an oscillator satisfying the initial value problem u?? + ?2 u = 0, u(0) = u0, u? (0) = ?0. (i) (a) Let x1 = u, x2 = u? , and transform Eqs. (i) into the form x? = Ax, x(0) = x0. (ii) (b) By using the series (17), show that eAt = I2 cos?t + Asin?t ? . (iii) (c) Find the solution of the initial value problem (ii).

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