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m238sp06t3 - Mathematics 238 test three due Friday 1 Given...

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Mathematics 238 test three due Friday, May 26, 2006 1 . Given the initial value problem y + 169 y = 0 and y (0) = 0 and y (0) = 1 find the solution function y ( t ) determine the amplitude and the period of the solution y ( t ). find the first two positive values at which y ( t ) attains a local maximum value. 2 . Given the initial value problem y + 10 y + 169 y = 0 and y (0) = 0 and y (0) = 1 find the solution function y ( t ) determine the first two positive values at which y ( t ) attains a local maximum value calculate the quasi-period of the solution (the t -difference between locations of adjacent local maximum values) and the percentage decrease between adjacent maximum values. 3 . Use the variation of parameters technique to show that the solution to the initial value problem y + 4 y = g ( t ) and y (0) = 0 and y (0) = 0 is given by the formula y ( t ) = 1 2 t 0 - g ( s ) sin 2 s ds cos 2 t + 1 2 t 0 g ( s ) cos 2 s ds sin 2 t Show that this formula is equivalent to the more concise y ( t ) = 1 2 t 0 g ( s ) sin 2( t - s ) ds 4 . Given the differential equation ( x 2 + 9) y + 2 xy + 4 y
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