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Unformatted text preview: Math 415 Quiz 6A The position of a spring-mass system satisfies the initial value problem u 00 + ku = 0 , u (0) = 2 , u (0) = v. If the period and amplitude of the resulting motion are observed to be and 3, respectively, determine the values of k and v . The general solution is C 1 cos( k t ) + C 2 sin( k t ), and with our initial conditions, it becomes u ( t ) = 2cos( k t ) + v k sin( k t ). To find the period and amplitude, we should write our solution in the form u ( t ) = R cos( t- ), where R is the amplitude and is related to the period, specifically the period is 2 / . But also, when rewriting this sort of expression the coefficient of the variable inside the trigonometric functions stays the same, so = k = 2, i.e. k = 4. R = q 2 2 + ( v k ) 2 = q 4 + v 2 4 , so R 2 = 9 = 4 + v 2 4 , or v = 20 = 2 5. Find all the eigenvalues and eigenfunctions of the following boundary value problem. Assume all eigenvalues are real....
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