850] 
ON LINEAR DIFFERENTIAL EQUATIONS. 
395 
which will in fact be a sum of m determinate series multiplied by y 0 , y lt y 2 > •••, y m -1 
respectively; say the form is 
V = 2/0^0 + 2/i^Ti + ••• + ym-iXm-i> 
where X 0 , X 1} ..., X m _ x are each of them a series of positive integer powers of x — a. 
Each of these series will be convergent for values of x sufficiently near to a, or say 
for points x within the domain of the point a ; and since y 0 , y x , ..., y in _ x are arbitrary, 
each series separately will satisfy the differential equation; and we have thus m 
particular integrals of the differential equation. 
3. Suppose next that x— a is a factor of p 0 , the point x—a is in this case 
said to be a singular point in regard to the differential equation. The foregoing pro- 
d m v d m+1 y 
cess of development fails, as leading to infinite values of , y x m+i > • • • j and we 
have to consider the developments of y which belong to the neighbourhood of the 
point x = a, or say to the domain of the singular point x = a. This has to be done 
separately in regard to each of the singular points, and among these we have, it may 
be, the point 00. To decide whether 00 is or is not a singular point, we may in the 
differential equation write x = l/£; and then transforming to this new variable, and 
throwing out any common factor, the coefficients of the transformed equation will be 
rational and integral functions of t, without any common factor; say these are 
P 0 , Pi, ..., P m ; if t is not a factor of P 0 , then 00 will be an ordinary point of the 
original equation; but if £ is a factor of P 0 , then 00 will be a singular point of the 
original equation. 
4. In considering the singular point x = a, we may, it is clear, transform the 
equation to this point as origin, and it is convenient to do this; supposing it done, 
we have an equation wherein x = 0 is a singular point, viz. p 0 contains x as a factor, 
and we have to consider the developments of y which belong to the domain of this 
singular point x = 0. 
It is convenient to change the form of the differential equation by dividing the 
whole equation by the first coefficient p 0 , and then expanding each of the quotients 
■P 1 , — 2 , ..., in a series of ascending powers of x. The new form is 
Po Po Po 
d m y d m ~ x y _ 
d^+P‘d^ + - + P™y- 0 ’ 
or say 
P(y)=(i> Pi, i) m y = o, 
where p 1} p 2 , •••, p m now denote each of them a series (finite or infinite) of integer 
powers of x, but containing only a finite number of negative powers of x. 
5. Such an equation frequently admits of a “ regular integral ” of the form 
y = x p E(cc), where E(cc) is a series of positive integer powers E 0 + E x x + E 2 x 2 +..., 
(E 0 not = 0, for this would imply a different value of p)*. To determine whether 
* The expression regular integral is afterwards used in a more general sense, see post, No. 10. 
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