Problems 52 and 53 please

48-53. Solution verification Verify by substitution that the following differential equations are satisfied by the given functions. Assume that C1 and c2 are arbitrary constants. lu- 48. y" - 12y' + 36y = 0; solution y(1) = Cled + Cate of ch 49. y" - 12y' + 36y = 26f. solution y = Clear + cate 6 + zeb 50. y" + 4y = 8 sin 2r; solution y = c sin 21 + C2 cos 2t - 2r cos 2t 51. 12y" - 3ty' + 4y = 0, 1 > 0; solution y = cit + C212 Int 52. 12y" - 3ty' + 4y = 212, 1 > 0; solution y = cit2 + cat? Int + 12 In2 53. 12 y" + ty' +. ( 12 - - ) y = 0,1> 0; rob- solution y = 1 1/2(c, cost + C2 sin t) 54. Trigonometric solutions a. Verify by substitution that y = sin / and y = cos / are solu- tions of the equation y" + y = 0. b. Write the general solution of y" + y = 0. c. Verify by substitution that y = sin 2/ and y = cos 21 are solu- tions of the equation y" + 4y = 0. d. Write the general solution of y" + 4y = 0. e. Based on the results of parts (a)-(d), find the general solu- tion of the equation y" + kay = 0, where k is a nonzero real number