Low Battery Your Mac will sleep soon unless plugged Find the general solution of the differential equation into a power outlet. x y" + xy' t (3x2 -9 )y = 0. OO Answer: y = c > Cnan , n=m where On = 0 if n > m and n = 2k is even, Cn if n > m and n = 2k + 1 is odd, m 3 , and c is an arbitrary constant. Choose your answer so that Cm = 1. Note: Because the coefficient of y" is zero when a = 0, you will not find two linearly independent solutions of the differential equation defined near a = 0. In fact, by looking at the differential equation, you can see that any solution defined on an open interval containing a = 0 must have y(0) = 0.Now, we have that Cn = Cn- 2 for n odd 9 -12 Also Cm = 1 1. e. ( 3 = 1 Then, C3 = 1 65 = C3 = 9- 25 - 16 (7= 65 = (-40) (76) = 9- 49 16 ( 40 ) Cq = CF = ( = ) (to ) ( to ) 9-81 which gives a general formula, Cn = (- 1) " ( 2) ( 8 ) ( 4 ) ( 10 ) ... ( 2n ) ( 24 + 6 ) Cn nam and n= 2k 4 even Cn ( - 1) ? 4 2 7 , m and n= 2k + 1 is odd ( 2 ) ( 8 ) ( 4 ) ( 10 ).... ( 2n ) ( 2n+ 6 ) m = 3 and C is an arbitrary constant or 3