460 Transactions of the South African Philosophical Society. 



§2. The first question we consider is — 

 Under what conditions can a solution of — 



^=a T y+a 2 y* (1) 



with initial condition y = \ when £ = 0, be obtained in a series of 

 ascending powers of X ? 



[This simple form is taken in order to illustrate the method used 

 in the more general case, and also because the result can be tested 

 by comparison with the solution in finite terms.] 



To determine the possibility assume a solution of the form — 



2/ = T + XT I + ^T 2 + (2) 



and determine the T's so as to satisfy the differential equation. If 

 the series thus obtained is convergent the existence of a solution of 

 this form is established. 



From the initial conditions we have — 



T o = 0, T T -1, T 2 = 0, T 3 = 0, ... when t = 0. 



Substitute from (2) in (1) — 



.". T;+XT' I +X 2 T;+...=a I (T +XT I +\-T 2 4-...) 

 + a 2 (T +XT I + X 2 T 2 ...)*, 



dV 

 where TJ. means -jj . 



Equating powers of X — 



T =a z T Q +a 2 Tl \ 



t; =^+^.2^ I 



T a = a/T 2 + a a (T; + T .T 2 )| ^ 



&c. J 



The first of these has the initial condition T o = when £ = 0; hence 

 initially T o = ; and thus T o = for all time. 

 Transform the set (3) by the substitution — 



T, = S r . £ ^ (4) 



and the equations (3) become a set of which a typical member is — 



s;.=a 2 . £ ^[s o s„+s I s r _ I +...+s r _ I .s I +s r s ] (5) 



The initial condition applicable to T x gives us S x = l when £ = 0. 



Further substitute — 



S r =^- lR r (6) 



m 



