FORMATION OF SERIES 111 



a series of terms of ascending powers of x. In other words, we 

 can solve a differential equation and give the result as a 

 series. 



Example. If -^ - 2~ + 2w = 0, and when x - 0, y = 0, and 

 ax* ax 



= 1, express y as a series of terms of ascending powers of x. 



CMP 



Then y = A^ + A^ + A^ 2 + A^ + A 4 ar* + A 6 a^ + . . . 

 -j| = A! + 2A2*: + SA^ 2 + 4A 4 ar } + SA^ 4 + . . . 



^ = 2A 2 + GAjjtf + 12A 4 * 2 + 20A 5 # 3 + . . . 

 When x = 0, y = 0, then AQ = ; and when x = 0, -r- 1, 



H0 



then A! = 1. 

 Also 2A 2 + 6A&+ 12A 4 a; 2 + 20A 6 3 + 



2A 4 a; 4 + . . . 

 Then 2A 2 - 2A! + 2A = A 2 = 1 



Since ^-2^ 



, I , * flflP 



6A 3 - 4A 2 + 2A X = 6A 3 = 2 A 3 



12A 4 - 6A 3 + 2A 2 = 12A 4 = A 4 = 



20A 6 -8A 4 + 2A 3 = 20A 5 = -| A 6 = - ^ 



30A fl - 10A, + 2A, = 30A rt = - - A, = - - 



3 90 



Hence y =* ec + x z + j: qj: etc. 



69. Sometimes, when it is necessary to expand a certain function 

 of x, it is convenient to form a simple relation between the first 

 and second differential coefficients and use this relation in the 

 above way. 



