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Mathematics 238 test three due Friday, May 26, 2006 1. Given the initial value problem y 00 + 169y = 0 and y(0) = 0 and y 0 (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 00 + 10y 0 + 169y = 0 and y(0) = 0 and y 0 (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 00 + 4y = g(t) and y(0) = 0 and y 0 (0) = 0 is given by the formula t t 1 1 −g(s) sin 2s ds cos 2t + g(s) cos 2s ds sin 2t 2 0 2 0 Show that this formula is equivalent to the more concise Z Z y(t) = y(t) = 4. Given the differential equation 1Z t g(s) sin 2(t − s) ds 2 0 (x2 + 9)y 00 + 2xy 0 + 4y = 0 • list the singular points of the equation. ∞ • for which x-values will a power series solution of the form X a (x−2)k converge? k k=0 5. For which values of x does the each of following power series converge? (a) ∞ X (x + 3)2k−1 k=1 (b) ∞ X k=1 k4k a2k x2k where a2 = 1 & ka2k+2 = −5(k + 1)a2k for k ≥ 1 page two 6. Find a power series solution about x = 0 (through the degree five term) to the secondorder initial value problem d2 y = −4 sin y and y(0) = 0 and y 0 (0) = 1 2 dt Compare your power series solution (through the degree five term) to the power series solution for the related problem d2 y = −4y dt2 and y(0) = 0 and y 0 (0) = 1 extra credit part of problem (6) • compare your solutions to these initial value problems graphically. Compare both solutions to a numerically derived solution of the first initial value problem, using some mathematical software) 7. Given the second-order differential equation y 00 + 2xy 0 + y = 0 • for which values of x will power series solutions of this differential equation converge? • find the recursion relation for the power series solution about x = 0. • write the general solution through the degree five terms. • find the solution (through degree five terms) to the initial value problem y(0) = 0 and y 0 (0) = 1. • estimate the accuracy of your solution of the previous part when −0.5 ≤ x ≤ 0.5. 8. Use the elimination method to solve the system of equations d2 x1 dt2 d2 x2 dt2 = x2 − x1 − 1 = −(x2 − x1 − 1) + (x3 − x2 − 1) d2 x3 dt2 = −(x3 − x2 − 1) Begin the process by rewriting the system in the format (D2 + 1)x1 − x2 = −1 −x1 + (D2 + 2)x2 − x3 = 0 −x2 + (D2 + 1)x3 = 1