Rit (Vísindafélag Íslendinga) - 01.06.1946, Side 11

11
It is to be observed that Table I is also applicable if
r>s. We have then only to interchange r and s in the
Table.
The working out of the solution of the differential equa-
tion from (10) or (11) may be difficult indeed, we are,
however, in many cases able, by means of successive ap-
proximations, to arrive at a sufficiently accurate solution
without laborious calculations.
2. LINEAR DIFFERENTIAL EQUATIONS
If the differential equation is linear the work of solution
may be performed in more different ways, and as this classe
of diff. equations in some respects may be considered as
the most important, it seems- worth while to indicate the
principal methods.
We write the diff. equation of order m as follows
— A(m) u + A(m) — +
dxm ~ A*’°y + dx+ • •
tfm)
dm~xy
dx
~\+9
,("i)
(14)
where A(m\, A(m\ .... and g(m) are known functions of x.
Hence by differentiation (n > m)
— = A(n) i/ + A(n) dlJ +
dxn *’oí/+
For x = 0 we have
,m—1
A(n) ________y- j_ 0(»)
1 dxn~' 9
(15)
dLyl = A(n)u +A(,,)^ +
dxn °’°y°+ °’1 dx + '
jm—1
I dW _______+°+0(n>
I ^O, m—1 ^ m-1 l vo
(16)