Viewing Saved Work Revert to Last Response 7. [-/3.5 Points] DETAILS SCALCET9 9.5.020.MI. Solve the initial-value problem. 13 dy + 3t2y = 2 cos(t), y(7) = 0 8. [-/3.5 Points] DETAILS SCALCET9 9.1.016. (a) For what values of k does the function y = cos(kt) satisfy the differential equation 81y "- -16y? (Enter your answers as a comma-separated list.) (b) For those values of k, verify that every member of the family of functions y - A sin(kt) + B cos(kt) is also a solution. We begin by calculating the following. y - A sin(kt) + B cos(kt) = y' = Ak cos(kt) - BK sin(kt) Note that the given differential equation 81y "= -16y is equivalent to 81y" + 16y Now, substituting the expressions for y and y" above and simplifying, we have LHS = 81y "+ 16y = 81 + 16(A sin(kt) + 8 cos(kt)) -81 81Bk2 cos(kt) + 16A sin(kt) + 168 cos(kt) = (16 - 81k2) + (16 - 81k2) B cos(kt) - 0 since for all value of k found above, k? = 23