Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 17

17
and I get a solution analogous to that of (23) where the
d"//0
coefficient of is a linear function of i],, i]2, etc. By
solving linear equations we are, therefore, able to find
such values of r\2, i]3 etc. as to render the coefficients
dk ~ xy0 dk ~ 2y0
of —-jfirr >----e*c- eciua* to zero. In practice we
dx dx
restrict the number of the unknown r| as far as possible.
Usually we content ourselves with iij or with r|, and n2.
In practical applications it is frequently necessary or
dy
advantageous to know — and possibly also other deriv-
dx
atives of y up to
dT^'y
dxm~
j Usually we do not find them by
differentiation of (23) because we prefer calculating them
in a similar way as y in (23) directly from the differential
equation by substitution from (19) where we successively
put p — 1, p = 2............m — 1.
We may also by means of the General Serial Relation
(3) arrive at the desired solution of the differential equation.
To this end we contemplate the linear diff. equation:
y = y + t ii/ + + ii2/'<'
f1
Tlr-l í
(r—1)
dy x d"y
= p0y-Pix— + p2~ ~T2 -+• •
dx 2! dx
= Q0y0+dix
dy0
dx
x
d2y0
f/22! dx2
(25)
Now /= 0 is a differential equation of order m and there-
fore in (25) pn = 0 for n > m + r — 1, whereas pn is a linear
function of r|0,rjj,ri2 etc. for n<m-\- r— 1 According to
(4b) also the coefficients q0, qv q2... are linear equations