I 11. a] Solve the second order linear differential equation 3:\" + 15y' + 143.! = e' . b] Briey explain howr you might obtain a unique solution, ifadditionally some initial conditions were also given in part [a]. b] Briey explain, what would be the answer ifthe auxiliary equation above had only one repeated root. (You may consider any of the roots you found in part {a]]. 13. a) Solve the second order linear differential equation y" + 16 y = 0. b) Write one particular solution, based on the general solution you obtained in part (a).True False Questions [3 points each] For questions 1-5, decide if the given statement is true or false, and briey justify for your answer. 1. If the auxiliary equation of a 2nd-order, homogeneous, linear differential equation with constant coefcients has roots r1 = 171' then the corresponding differential equation is y" + 73: = 0. If the auxiliary equation of a 2nd-order, homogeneous, linear differential equation with constant coefcients has one repeated real root r1 = r2 = 5 then the general solution is: y(x) = c1 cos (5x) + 2 sin(5;x). If 31,, is a particular solution to the differential equation 3!" + mg.\" + azy = 52:, then 3!? + 4-11 is also a solution to y\" + aly' + azy = 53: for every solution 1: ofthe corresponding homogeneous differential equation 3:\" + a1y' + 0.231 = 0. If the auxiliary equation of a 2nd-order, homogeneous, linear differential equation with constant coefficients has two distinct real roots 1 and 8 then the general solution is 31(1) = clex + r:2 93". If 3:1 is a solution to y\" 123: = F1 and y; is a solution to y\" 123: = F2, then 3y1 + 23!; is a solution to y\" 123; = F1 + F2