(8) w7.4) Consider the configuration above, in the undamped case c = 0. In particular consider the IVP x" +.64x = 2 cos(

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8 W7 4 Consider The Configuration Above In The Undamped Case C 0 In Particular Consider The Ivp X 64x 2 Cos 1 (159.05 KiB) Viewed 29 times

(8) w7.4) Consider the configuration above, in the undamped case c = 0. In particular consider the IVP x" +.64x = 2 cos(wt) x(0) = 0 x' (0) = 0 (a) What is the "natural" angular frequency wo (for the unforced problem) in this differential equation? Hint, the natural frequency is defined to be the angular frequency for the solutions to the unforced and undamped differential equation, which in this case is the DE x" + .64x = 0 (b) Assume w wo: Use the method of undetermined coefficients to solve for the particular solution xp(t) for the forced differential equation. Then use x(t) = xp(t) + xH(t) to solve the IVP. Check your answer with technology. = (c) Write down the special case of the solution in (b) when w = 0.7 Computer the period of this solution, which is the superposition of two cosine functions. Use technology to graph on period of the solution. What phenomenon is exhibited by this solution? (d) Solve the IVP when w = wo. Use the method of undetermined coefficients or operator factorization i.e. L = D² + .64I [D+.8iI] o [D.8iI] as in w7.2, to find a particular solution, and then use x = xp+xH to solve the IVP. Check your answer with technology. Graph the solution on the interval 0 ≤ t ≤ 60 seconds. When phenomenon is exhibited by this solution