1. [-/2 Points] DETAILS SCALC9 9.1.015.MI. MY NOTES ASK YOUR TEACHER (a) For what values of r does the function y = efx satisfy the differential equation 3y" + 14y' - 5y = 0? (Enter your answers as a comma-separated list.) r = (b) If ri and r2 are the values of r that you found in part (a), show that every member of the family of functions y = aeli* + be/2* is also a solution. (Let /j be the larger value and /2 be the smaller value.) @ Show My Work (Optional)? 2. [-/7 Points] DETAILS SCALC9 9.1.016. MY NOTES ASK YOUR TEACHER (a) For what values of k does the function y = cos(kt) satisfy the differential equation 81y" = -49y? (Enter your answers as a comma-separated list.) K = (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. y = A sin(kt) + B cos(kt) = y' = Ak cos(kt) - Bk sin(kt) = y" = -Ak2 sin(kt) - BK2 cos(kt). The given differential equation 81y" = -49y is equivalent to 81y" + 49y = Thus, LHS = 81y" + 49y= 81(-Ak2 sin(kt) - Bk2 cos(kt)) + 49( = -81Ak2 sin(kt) - 81Bk2 cos(kt) + sin( kt ) + 49B cos(kt) = (49 - 81k2)A sin(kt) + cos ( kt ) since k2 = Show My Work (Optional)