USE
OF
POWER
NONLINEAR
IN S O L V I N G
ORDINARY
AND SYSTEMS P.F.
SERIES
LINEAR
DIFFERENTIAL
AND
EQUATIONS
THEREOF
Fil~chakov
UDC 517.912
1. F o r the solution of a b r o a d class of linear and nonlinear ordinary differential equations, we shall make use in this paper of power s e r i e s , the possibilities of which have still been inadequately exploited in the l i t e r a t u r e . The use of power s e r i e s for the integration of differential equations had its beginnings simultaneously with the development of the bases of differential and integral calculus. In a number of papers and m e m o i r s of Isaac Newton, G. W. Leibniz, John Bernoulli, and J a m e s Bernoulli a s y s t e m a t i c exposition was given of the method of undetermined eoeffieients for the solution of linear and some other differential equations. A f u r t h e r large step in this direction was taken by Leonardo Euler. Euler p r o p o s e d to seek the solution of the equation y ' = f(x, y), with initial condition y(0) = 0, in the f o r m of a s e r i e s , y = aix +a2x 2 +a3x3 § . . . . whose coefficients were to be determined in a c c o r d a n c e with the T a y l o r f o r m u l a
an =
y(n) (0). n!
I
,
a, = y' (0) = : (x, y) ~=00 ,.
All the derivatives, y(n), n e c e s s a r y for this are found by s u c c e s s i v e l y differentiating the initial equation: y,,
o:
of
=-~-+-0Vv';'
v"
~
,~ o~f
,
=~+ ~y
o~: :
of ,,;
+07y +-~-~ ...
Euler used this method 200 y e a r s ago, and explained it in the second volume of his m e m o i r s , Institutions Calculi Integralis, Petropoli, 1769. Subsequently, E u l e r ' s method was widely used in the solution of various theoretical questions, but its p r a c t i c a l usage was hampered by the increasing complexity of differentiating function f(x, y) as n increases so that, ordinarily, it is difficult to find m o r e than five to ten t e r m s of the s e r i e s using this method. All these difficulties are r e m o v e d if one finds the coefficients an, not by T a y l o r ' s formula., but r a t h e r by Cauc h y ' s f o r m u l a for the multiplication of power s e r i e s . We also r e m a r k that Cauchy,s f o r m u l a p e r m i t s , for a v e r y b r o a d class of differential equations, the development of r e c u r s i v e algorithms which are easily p r o g r a m m e d for c o m p u t e r s . 2. Consider the Cauchy p r o b l e m for the s y s t e m of o r d i n a r y differential equations
v ( • J")(x,i
-- li
YI' g/l, '
....
if1-')
, ..
., v,, v, ' .....
(i)
. v,(':r--l) )
with initial conditions v~ (x0) = v,0;
y)(x0) = rio; ... ; yl `:i-') (Xo) = Vloi-'),
i = 1, 2 .....
r.
(2)
We r e q u i r e of the functions fj that they be analytic in all their arguments. We now introduce the notation which we shall need in the sequel. We denote the coefficients of the series r
(3) n~0
n=0
n~0
Mathematics Institute of the Ukrainian SSR. T r a n s l a t e d f r o m Ukrainskii Matematicheskii Zhurnal, Vol. 21, No. 2, pp. 230-237, M a r c h - A p r i l , 1969. Original a r t i c l e submitted November 28, 1968.
182
equal to the p r o d u c t of two s e r i e s , by the s y m b o l c,, = [A,,B,,],
the n u m e r i c a l value of which is d e t e r m i n e d by C a u e h y ' s formula, [A B ] = Ao/~,, + AIB~_ , + A~B~_ 2 -~ .. . § A Bo,
(4)
n = O, t, 2 . . . .
In this c a s e , if we s e e k the solution to the p r o b l e m of (1), (2) in the f o r m of the p o w e r s e r i e s
(5)
gi ---- ~ ain (X ~ )Co)n, r~=0
then any i n t e g r a l p o w e r of yj can a l s o be p r e s e n t e d b y a p o w e r s e r i e s
(6) r~=0
w h o s e c o e f f i c i e n t s , a~nk), a r e e a s i l y c o m p u t e d by formula. (4). Indeed, by s e t t i n g
in (3), we a r r i v e at the r e c u r s i o n f o r m u l a a(.l+m) ~ ail)a(m) 27 a(t)a (m) ,.. ~n iO in i! i,n--! - ~
~
w h e r e l = 1, 2 , 3 . . . . ; m = 1, 2 , 3 . . . . ; n = 0 , 1 , 2 . . . . ; j = 1,2 . . . . . i d e n t i c a l l y with s e r i e s (5) when k = 1, s o thas a .(1)In~
ajn,
] = 1, 2 . . . .
r;.
(7)
g(l)a.(m) in 70
r. Whence, s i n c e s e r i e s (6) coincides
n . = 0,. 1, 2,
,
(8)
we d e t e r m i n e , in a c c o r d a n c e with f o r m u l a (7), the s u c c e s s i v e c o e f f i c i e n t s a! k) f o r k = 2, 3, 4 . . . . . only the c o e f f i c i e n t s , ajn , of the o r i g i n a l s e r i e s in (5). F o r c o n v e n i e n c e in computing, we p r e s e n t the d e r i v a t i v e s yTj, y , j . . . . series:
n~O
using
in the f o r m of the following
n~O
w h e r e we u s e the notation a i : + l = ( n + 1) ai.n+~; ai,,+ 2 = (n + 1)(n + 2)al.~+~;...
(10)
In this c a s e , we can a l s o e a s i l y e x p r e s s all the p o w e r s of the d e r i v a t i v e s , y,j, y , j . . . . . (y;)k=
..... -
(y;;:=
.,
n~O
w h o s e c o e f f i c i e n t s h (k) j , n + ~ ; .~(k) j,n+2;
by s e r i e s
n~0
...(k) aj,n+a;""
a r e d e t e r m i n e d b y m e a n s of forn'mla (7) if, in t h i s f o r m u .
la, we r e p l a c e the ajn by, r e s p e c t i v e l y , a j , n + 1; k j , n + 2; " a j , n + a ; - . If we now s u b s t i t u t e the s e r i e s of (5), (6), (9), and (11) into the s y s t e m of equations of (1), and then equas c o e f f i c i e n t s of identical p o w e r s of ( x - x o ) n, we obtain, a f t e r the a p p r o p r i a t e s i m p l i f i c a t i o n , r e c u r s i r e f o r m u l a s of the f o r m ai,,~+,~i =Fi(aio;ai~;... ;ai,n+vi_~);
n ~ 0, I, 2 . . . . .
(12)
w h e r e the functions Fj a r e c o m p l e t e l y defined by the r i g h t sides of the equations in s y s t e m (1) and w h e r e the f i r s t pj c o e f f i c i e n t s , in a c c o r d a n c e with the in[tiM conditions of (2), will be: 9
aio=Ym;
ai~=
Ii '
.
ai2=~
.... ;
:7/'0
aJ'vi-l -- (v i - l ) [ "
(13)
183
In c a s e s w h e r e t h e l i m i t
ain
lim
n..-).vo a i.,n+p
e x i s t s , the r a d i u s of c o n v e r g e n c e of s e r i e s (5) c a n b e d e t e r m i n e d n u m e r i c a l l y b y c o m p u t i n g a s u f f i c i e n t n u m b e r of t e r m s of the s e q u e n c e R~=
ai~
,
n=0,
1, 2 . . . . ; p = : l, 2, 3 . . . .
(14)
al,n+p
up to the v a l u e of n with w h i c h b e g i n s the holding of the e q u a t i o n s ain = a i ' n + ~ = at'n+~-----L. . . . Cli,n+p ai,n+2p ai,n+3p
= const = R p
(15)
to w i t h i n the a c c u r a c y d i c t a t e d b y the g i v e n p r o b l e m . T h e m e t h o d we have b e e n c o n s i d e r i n g , w h i c h a l s o r e m a i n s v a l i d in the c o m p l e x d o m a i n , a l s o a l l o w s us to d e t e r m i n e the a n a l y t i c c o n t i n u a t i o n of s e r i e s (5), and to e x h i b i t the s i n g u l a x p o i n t s o f ' t h e s o l u t i o n s we find. We c a n e l u c i d a t e a l l t h e s e q u e s t i o n s b e t t e r b y c o n s i d e r i n g e x a m p l e s , the f i r s t of wh:tch t r e a t s the c a s e of a s i n g l e e q u a t i o n (j = 1). E x a m p l e 1. We s h a l l s o l v e the C a u c h y p r o b l e m f o r the R i c c a t i e q u a t i o n :
y'--y2+~(x);
V(0)=l;
~ ( x ) = x 2.
(16)
By s u b s t i t u t i n g into Eq. (16) the s e r i e s of (5), (6), and (9) f o r x 0 = 0, j = 1, and k = 2, t a k i n g into a c c o u n t t h a t , a c c o r d i n g to the i n i t i a l c o n d i t i o n s , a 0 = y(0) = 1, we find, a s the r e s u l t of e q u a t i n g l i k e p o w e r s of x, that
an+ l=a~ 2)+Y~;
a o= I,
(17)
w h e r e the ~/n a r e the T a y l o r s e r i e s c o e f f i c i e n t s f o r the g i v e n f u n c t i o n ~(x) and, in the g i v e n c a s e , 3/2 = 1, 7n = 0 when n ~ 2. F o r the s p e c i a l c a s e of a s i n g l e e q u a t i o n , we s h a l l o m i t the f i r s t s u b s c r i p t , j = 1, in w r i t i n g the c o e f f i c i e n t s , a j n = a n , of the s e r i e s (5) we s e e k .
TABLE
[84
1
n
an
a~)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 t8 19 20 21 22 23
1,0000000 1,0000000 1,0000000 1,3333333
1,0000000 2,0000000 3,0000000 4,6656667 6,0000000 7,4000000 8,9777778 10,5428572 12 2285714 14:0071428 15,8924858 17,8753102 19,9676767 22,1727752 24,4963402 26,9427497 29,5177952 32,227O317 35,0763686 38,0718547 41,2198479 44,5259358 47,9999759
1,1666667 1,2000000 1,2333333 1,2825397 1,3178571 1,3587302 1,4007143 1,4447715 1,4896092 1,5359751 1,5837697 1,6330893 1,6839219 1,7363409 1,7903906 1,8461247 ,1,9035927 1,9628499 2,0239516 2,0869555
~n
0
0 o 0 0 0 o 0 0 0
R n - - aan~ - -
1,00000000 1.,000000O0 0,75000000 1,14285708 0,97222222 0,97297297 0,96163362 0,97320089 0,96991816 0,97002665 0,96950577 0,96989969
0,96981338 0,96982225 0,96979981 0,96981297 0,96981065 0,96981122 0,96981022 0,96981077 0,96981063 0,96981069 0,96981062
bn
+0,0311292 +0 03'11292 --0:3022041 q:0,2081722 +0,OO29841 +0,0040217 --0,0108137 40,0046070 +0,0001507 +0,0003121 --0,0004541 +0,O001369
+0,O000044
+0,0000191 --0,0000181 -}-0,0000042
+0,0000001
+}-0,0000012 --0,0000007 +0;0000004
+0,0000001 +0,0000002 0,0000000
in (17) an+ i = (n+l)an+ l according
If we now make the replacement eursion formulas of (12) for Eq. (16)
a(n2) + Y" n+l ;
a~+l=
ao=t;
to formula
(I0), we find the re-
I~ for n = 2, for n 4 = 2 .
~=,~
(18)
W e c o m p u t e t h e c o e f f i c i e n t s a n (2) f o r the s e r i e s y2 = E a n ( 2 ) x n b y f o r m u l a (7), in which we must l =m=l: a n(2)
= aoan
+
-1- a l a n _ l
a2an_ 2
In Table 1 we give all the computations 24 coefficients a n of the solution to Eq. (16)
+.
9
necessary
9
+ a,~ao,
set
(19)
n = O, 1, 2 . . . .
for determining,
with 7-place accuracy,
the first
a n X ~.
y =
(20)
S i n c e c o n d i t i o n (15) is f u l f i l l e d f o r the c o e f f i c i e n t s a n we h a v e f o u n d w h e n p = 1, w e c a n , b y t h i s s m ~ e table, compute the quantities = ~ =
R,,
and the corresponding
q,2
;
n~.o
coefficients b n of the series ;
obtained by computing,
(21)
R=lim
an+!
from
R = 0,9698106,
the original series (20),the geometric - -ao 1 - - qx
-
a,
{ 1 + qx + q2x2 + qSx3 + . . .
progression
(22)
with denominator
q = l/R:
}.
(23)
Generally speaking, the quantity q may be real or complex. We compute the coefficients bn by the formula bn+ I = ar.q--an+l;
which
is easily obtained by using the formula
b 0 = i,
(24)
for the division of series ([13], w 16).
Thus, the first singular point of the solution we seek for Eq. (16) is the (simple) pole x = +R, axld series (22), in which this pole appears, allows us to carry the computations beyond the limit thus imposed. Thus, for example, when x = i, we have, according to (22), y(1) =
R(1 + 0.0308179) = _ R--1
33.1142,
w i t h t h e e x a c t v a l u e at p o i n t x = 1 b e i n g g (1) = - - 33.1143556...
Computing, by means of series (22), a new initial value, Y(X0) = Y0, beyond the limits of the first singular point, i.e., beyond the radius of convergence, R, of the initial series (20), we determine analogously the second singular point. By continuing this process, we can continue analytically the inltia.l element of the solution of (20) to any previously specified finite interval. In winding up the treatment of Example i, we note that the course of solu.tion remains the same for any other analytic function, (fl(x). Thus, if ~(x) = cos x, it is necessary to substitute in column Yn of Table 1 its new values: Y2v=
(-- ly. (2v)l '
Y~v+l=0, v=0,
1, 2 . . . .
Raising the order of the equation leads to no difficulties in principle and, for example,
for the equation
y " = g~ + r (x)
188
w e h a v e , i n s t e a d of the r e c u r s i o n f o r m u l a s of (17) and (18), 9.. a._~.3 = a(.:) + Yn;
a~) + ~, an+a = (n q- 1) (n + 2) (n + 3)"
E x a m p l e 2. We s o l v e the Ca.uchy p r o b l e m f o r the e q u a t i o n r (x) = ex = ~ "~nxn
4g" = g'%]- sin g q- r (x);
(25)
n=0
with initial conditions
x0 = 0 ;
y'(0) = g " ( 0 ) = - ~1.
g(0) --- I;
By e x p a n d i n g s i n y in a M a c l a u r i n s e r i e s ,
(26)
w e p u t Eq. (25) in the f o r m co
4g" =
y--~-!
+ 5t
(27)
7r + " "" +
In this c a s e , the r e c u r s t o n f o r m u l a of (12) w i l l b e t h e f o r m u l a .
"/
a. - - ~
{
(3, a2, a,,,
4a,,+3 = or
a~ a~-- ~-b
a~+3 ----"
+
5!
1
"" " +
a(~t
+ Y~
(28)
}
a ~2~ + V. 5! 71 t - . . . q- ~+l 4 (n q- 1) (n q- 2) (n -q- 3) ' n=O,
where,
7! +
1 Y" = ~ ;
(29)
1, 2 . . . . .
a c c o r d i n g to the i n i t i a l v a l u e s of (26), ao = 1 ;
A l l the c o m p u t a t i o n s n e e d e d to d e t e r m i n e , s o l u t i o n y = Z a n xn a r e g i v e n in T a b l e 2 w h e r e , s t a r t i n g w i t h n >-- 6, the a d d i t i o n a l t e r m s an(13), T h u s , in t h i s e x a m p l e we have t a k e n a c c o u n t of
a,=-~;
1
I a2=-~-.
with 7 - p l a c e a c c u r a c y , the f i r s t 16 c o e f f i c i e n t s of the in o r d e r to g u a r a n t e e the r e q u i s i t e a c c u r a c y , we c o m p u t e d , a n (l@, a n (17), and an(19), whieh a r e not shown in the t a b l e . the n o n l i n e a r i t i e s of the f u n c t i o n s i n y up to the 1 9 - t h p o w e r
inclusive. W e note t h a t the c o e f f i c i e n t s a n (2) and an+ 1 = (n+l)an+t, w h i c h a~'e n e c e s s a r y f o r the c o m p u t a t i o n of an(2V+l), u = 1, 2, 3 . . . . . and &(n2+)1in the r e c u r s i o n f o r m u l a of (29), a r e not u s e d d i r e c t l y . T o c o m p u t e the n e x t an+ 3 a c c o r d i n g to (29), it is n e c e s s a r y to m u l t i p l y e a c h of the a n (2v+l) b y t h e c o e f f i c i e n t (--1) v + l / ( 2 v + l ) ! ( w h o s e v a l u e is to b e found in the top r o w of T a b l e 2) and t h e n s u m the r e s u l t s . E q u a t i o n (25) c a n b e s o l v e d m o r e s i m p l y b y b r i n g i n g it to an e q u i v a l e n t s y s t e m of e q u a t i o n s with a l g e b r a i c r i g h t m e m b e r s . T o this end we i n t r o d u c e two a u x i l i a r y f u n c t i o n s z =sing;
o=cosg.
(30)
T h e n , t a k i n g into a c c o u n t t h a t z' = y' cos g = y'v;
v' = - - y'sin y = - - y'z,
w e a r r i v e a t the r e q u i r e d s y s t e m of e q u a t i o n s 4y'"=g'2+z+e~;
186
y(O)=l;
1 g'(O)=y"(O)=-~;
z' = + g'v;
z (0) = sin go = sin 1 = 0.8414710;
v' = - - g'z;
v (0) = cos go = cos 1 = 0.5403023.
(31)
0,3009535.10--~ 0,6410699.10- s 0,1789073.10-5 0,6251691.I0--6 0,2308640.10-6
8
15
II 12 13 14
I0
9
7
5 6
3 4
2
0,8035061.10--7 0,2529085.10--7 0,7046618.10--8
1,0000000 0,5000000 0,2500000 0,8714462.10- I 0,1843907.10- I 0,4338857.10-2 0,9010440.1043 0,1640162.10_3
0
I
an
n
4.~o~g '~-{-sin y-}-ex;
TABLE 2
0,1609736
0,1056773.10-3 0,2535767.10-4 0,6443776.10--5
0,66t2236.10 -2 0,1770944.10-2 0,4419761.10-3
0,3581592 0,6415900.10-1 0,2312603.10~1 0,7635440.10-2 0,2344021.10_2 0,6788172.10-3 0,1882105.10-3
1,5000000 1,1364338 0,6917510
0,2295466.10- I
1,0000000 1,5000000
a(3)
~I0 - I 1,66666667
1,0000000 0,7500000 0,4242892 0,1865228 0,7068909.10-1
1,0000000
a(2n)
(--l)'V'i- !~ (2v-}-l) t
1,25000 ~0,28430
1,00000 4,50000
a(9)
_~_i0--5 0,2755732
0,0227072t 0,579143
0,6154604i 5,538870 0,3019874t 3,527155 O, 1367075/ 2,072876 0,0575626] 1,132705
1,92889511 0,325239 1,1463032[ 7,954047
i
0,5000000 16,5000 0,2614339 35,3336 0,7375628.10 - I 59,9958 0,2169428.10- l 85,3988 0,5406264.10-2
5,5000
~4,82026 :9,45115 ~2,77582 6,25608 0,78954 6,70099
15,5747 14,3196 03,3224 86, t544 56,7924 48,4501
0,6876860.10-5 0,2770368.10-5 0,I044558.10 -5
0,2407628.t0 - a 0,5769629.10-4 0,1789073.10-4
a(:.~l
Y~
0,500000000
1,000000000
1,000000000
Yn
0,007415816 0,002166904 0,000605405 0,000169493 0,000050373 0,000016474
0,000000002
0,000000025
0,000002756 0,000000276
0,000198413 0,000024802
0,02333761 0,001388889
0,166666667 0,041666667 0,06566533 0,008333333
0,3351902 0,1637982
0,2500000 0,5000000 0,5114339
u Yn-~"I~ l ;
artJFl=(n-{-l) at;tq_1
t, 0000 [ 0,5000000
a(n11)
~I0--7 0,2505211
17,19273 05,4463 0,1148113.10-2
4,18572301 0,235012 3,7761414[ 2,021610 19,17816 2,8891426/ 1,932657 15,32407
2,5000000t 3,500000 3,7500000[ 7,000000
1,0000000 1,000000
.}-I0- 2 _10--3 0,83333333 0,19841270
0
o
2 3 4 5 6 7 8 9 10 11 12 13
0 -{-0,1666667 +o,0416667 @0,0041667 +0,0004167 +0,0000198 +0,0000186 --0,0000060 --0,0000037 -{-0,6000014 +0,0000004 --0,0000001'
1 +I,0000000
an
a
o +1,0000o00 o --0,1333333 -}-0,0416667 --0,0388333 --0,0370833 +0,021565o +0,0094353 --0,0035187
bn
@1,0o0o0o0 o --0,5000000 o -i-0,0083333 --0,0416667 i }-O,03allll -{-0,0218056 ~--0,0105900
Cll dn Yn
48 240 720 1680 3360 6048 10080 15840 23760 34320
0 -{-0,5000000 -{-{-0,1666667 F0,0208333 +0,0025000 [-0,0001386 @0,0001488 --0,0000540 --0,0000370
i+1,ooooooo
t
an+1
@0,8414710 +0,2701512 +0,2989170.10 - t --0,6935574.10 -1 --0,6769126.10 -1 --0,3355051.10 -~ --0,I180253.10 -1 --0,2461697.10 - 2 @0,2839639.10 - 3 @0,6108010.10 - 3 +0,3817575,10 - 3 .}.0,1705428.10 - 3 .}.0,6047250.10 - 4
I 2Nn
0,2500000 0,5000000 0,5114339 0,3551902 0,1637982 0,6566533.19 - I 0,2333761.10 - l 0,7415815.10 - 2 0,2166904.10 - 2 0,6054052.I0 - a 0,1694933.10 - 3 0,5037300.10 - 4 0,1647450.10 -4
ri~-i
,i(~.),
+1,0000000 ~__! 0 --0,4000000 0 +0,0266667 0 --0,0333333 -}-0,0346222 +0,0207778 ~--0,0085954
0,5000000 0,5000000 0,2614339 0,7375628.10 - I 0,2169428.10 -~ 0,5406264.10 - 2 0,1148113.10 - 2 0,2407628.10 _3 0,5769629.10 - 4 0,1789073.10 - 4 0,6876860.10 - 5 0,2770368.10 - 5 0,1044558.10 - 5
1,o0o0000 0,5000000 0,2500000 0,8714462.10 -1 0,1843907.1o -1 0,4338857.1o - 2 0,9010440.10 - 3 0,1640162.1o - 3 o,3oo9535.10 - 4 0,6410699.10 - 5 0,1789073.10 - 5 0,6251691.10 - 6 0,2308640.10 - 6 0,8035058.10 - 7 0,2529085.10 - 7 0,7046620.10 - 8
TABLE 4
10 11 12 13 14 15
8 9
5 6 7
4
2 3
1
0
an-}-1
at2
TABLE 3
0 @1,0003000 0 --0,6333333 +0,0416667 @0,03w --0,0995834 -{-0,O739816
[ bn Cn ]
0 @1,0000000 0 --0,5333333 -{-0,0416667 -}-0,0411667 --0,0870833 @0,0681650
[ Cn dn ]
I
r
@0,2701512 +0,05978340 --0,2080672 --0,2707650 --9,1677526 --0,07081521 --0,01723188 4.0,002271711 @0,005497209 +0,003817575 -}-0,001875971 -I-0,0007256700
[ ~,,+, ~,, 1
o @1,oo0000o 0 --0,0333333 -}-0,2083333 --0,2046667 --0,1526389 -}-0,0847203
@1,0000000 0 --0,400O000 -}-0,1666667 --0,I941667 --0,2225000 -}-}-0,1509552 @0,0754822 --0,0316683
[[b~ a. l a~+d [[ct2 at2] ~+z]
-}-0,5403023 --0,4207355 --0,2779056 --0,1233368 --0,02823961 -}-0,004505668 +0,009356363 +0,006059091 -I-0,002715784 -}-0,0009271740 @0,0002202300 -}-0,00001186817
0 @1,0000000 0 --0,1333333 -{-0,2083333 --0,2596667 --0,1818056 @0,0859536
q-0,4207355 -}-0,5558111 +0,3700105 +0,1129584 --0,02252834 --0,05613318 --0,04241364 --0,02172627 --0,008344566 --0,002202300 --0,0001305499
[ ~+, bn]
+1,0000000 0 --0,9000000 0 @0,2350000 --0,0750000 0,0520666 @0,0759167 --0,0499187
L
24 96 240 480 840 1344 2016 2880 3960 5280 6864 8736 0920
4Na
[ bn d n ]
l,OOO00000 1,00000000 0,50000000 0,16666667 0,41666667.10 -~ 0,83333333.10 - 2 0,13888889.10 - ~ 0,19841270.10 - a 0,24801587.10 - 4 0,27557319.10 - 5 0,27557319.10 - 6 0,25052108.10 - 7 0,20876758.10 - 8
'~'n
By now e x p r e s s i n g the t a r g e t functions, y, z, and v, and the given function, q)(x) = ex, by s e r i e s with one and the s a m e e e n ; e r ,
J~
/
X0 = 0, y=
anx ;
z=
n~O
b x~;
v = ~
n=O
nx ,
n~O
i" / /
o~
/ /
/
J
eX =
"~nxr';
Yn ~
nl
rt
'
n=O
we obtain the following r e e u r s i o n f o r m u l a s f o r d e t e r m i n i n g the c o e f f i c i e n t s an, bn, and en:
3s "T'-r-1
-1
I I I '
~ r J
§ Fig. 1
a , t 2• )b. +1
c
a'~+a =
+ y,~
4Nn
'
[a~+lC~] .
(a2)
b.+~-- n~- 1 " --
[an+ibn] n 4- 1
cn+l =
'
where ao = 1 ;
1
1
a~=~;
b0=sinl;
as=-4-;
Co=COSl;
N, = (n 4- 1) (n 4- 2) (n 4- 3). All the c o m p u t a t i o n s up to n = 15 inclusive a r e shown, with 7 - p l a c e a c c u r a c y , in T a M e 3. With this, we c o m p u t e the quantities [hn+ 1 bn] o r [an+t en] by formula. (4), s u b s t i t u t i n g into the f o r m u l a , c o r r e s p o n d ingly, An = hn+l; Bn = bn or B n = Cn. C o m p a r i n g the r e s u l t s of T a b l e s 2 and 3, we c a n c o n v i n c e o u r s e l v e s that t h e y c o i n e i d e w i t h i n the l i m i t s of a c c u r a c y of the c o m p u t a t i o n s . F r o m the c o e f f i c i e n t s a n and an+t thus found, one can e a s i l y c o m p u t e the value of the t a r g e t function, y(x), and its f i r s t d e r i v a t i v e , y'(x); on Fig, 1 we have c o n s t r u c t e d t h e i r g r a p h s on the s e g m e n t [--1, +1]. F o r l a r g e r (by modulus) values of x, one eeaa u s e a n a l y t i c continuation, f i r s t computing, f o r the new c e n t e r of the s e r i e s , x = x0, the n e c e s s a r y ~.nitial value, t h e r e a f t e r d e t e r m i n i n g the c o e f f i c i e n t s of this s e r i e s . E x a m p l e 3. We i n t e g r a t e the equation 2g w = g
1
1---~x;
+sn(g,k)+
ks = 0 ' 8
(33)
with the initial conditions xo = O ;
y (O) = y" (O) = O; y ' ( O ) = g " ( O ) = + l .
(34)
By putting the t a r g e t function in the f o r m of a s e r i e s y ~
anx
n
rt=O
and i n t r o d u c i n g the a u x i l i a r y functions
z=sn(g,k)=~
b.x~; v = c n ( y , k ) = s
c~x'; w = dn (y, k) = ~ d~x~ r~O
we a r r i v e at a s y s t e m which is equivalent to Eq. (33)
189
14 15 16
l
I
I
1
n
+I,000000o 0 --0,5000000 0 +0,0416667 0 --0,0013889 0 +0,0000248 0 --0,0000003 0 0 0 0 0
+I,0000000 0 --0,5000000 0 +0,0833333 0 --0,0125000 0 +0,0013641 0 --0,0001312 0 +0,0000111 0 --0,0000008 0
+0,00000006
Yn
--[Yn(n~-2) an] w"=--w cos z; an+2: (n--~l) an
TABLE 5
+1,0000000 0 ~1,0000000 0 +0,37500O0 0 --0,0763889 0 -{-O,0118O56 0 --0,0014625 0 -[-0,0001531 0 --0,0O00140 0
[Yn an]
E
8 12 16 20 24 28 32 36
0 4
n
wl(z)
j~z)
+ I, 242718
+3,663798 ~3,197746 +0,888038 ~0,120567 -{-0,009692 ~0,000515 +0,000019 --0,OO0001
Anq_3(z)
0. 000072899 Z~=Zlq - 3,370579
wx--wo=--O,O00072890;
--0,296604418
+I,0000000O0 --2,036522250 +0,888733473 --0,164538681 +0,016754334 --0,001077492 +0,000047703 --0,000001542 +0,0OO000038 --0,000000001
A n (z)
wo(z)
--0,296531528
--0,000000021
+0,000000926
--2,223400000 +2,716802142 --0,922138002 +0,144458043 --0,012979060 +0,0O0755204 --0,000030760
"
Anq_l(Z)
2.223378375
w : - - w~=+3,370579
J
z~zl=--2.2234; z~=24.43826700
~'o(z)
--2,127861
+ I, 000000 ~6,109567 +3,734208 --0,844978 +0,099278 --0,007136 +0,000345 --0,000012
A~nq_l(Z)
2glV=g'q-zq-(l---~x); z' = + vwg';
v' = - - zwy';
2
w' = - - k%vy'.
(as)
We d e t e r m i n e the c o e f f i c i e n t s a n , bn, en, and dn f o r the s o l u t i o n of s y s t e m (35) b y the f o l l o w i n g r e c u r s ion f o r m u l a s :
an+4 :
an+ 1 -[- b,~ + y,~ 2Nn ; b'~+l
[[Cndn] a~+ll n -}- 1 ;
(36) c+~=
[[b d ] a~+ll . n+ 1 '
d~+l=
k 2 [[b~cJ a~+ll n+l '
where, a c c o r d i n g to (33) and (34), Nn = (n + l)(n + 2)(n + 3)(n + 4); Y2ao=a2=0;
1 5 ; a a = I;
Yo=-}-l;
Yn = 0 when n = 1, 3, 4, 5 . . . . ; 1 an=-6-;
bo = 0 ;
co = d o = 1; /~2 = 0.8.
A l l the c o m p u t a t i o n s n e c e s s a r y f o r the d e t e r m i n a t i o n of the f i r s t 14 c o e f f i c i e n t s a n and the c o r r e s p o n d i n g b n , Cn, and d n a r e g i v e n in T a b l e 4. F o r t h i s , we f i r s t find, b y f o r m u l a (4), the q u a n t i t i e s IbnenI , [bndn[ , Icndnl, and then, s e t t i n g in t h i s s a m e f o r m u l a A =[c~d~l;
B =a+~
a n d d i v i d i n g t h e r e s u l t b y (n+l), we c o m p u t e t h e n e x t c o e f f i c i e n t , bn+ > C o e f f i c i e n t s c n and d n a r e found analogously, T h e m e t h o d we have b e e n c o n s i d e r i n g c a n a l s o be u s e d f o r the s o l u t i o n of b o u n d a r y - v a l u e p r o b l e m s and e i g e n v a l u e p r o b l e m s . In the c a s e of e q u a t i o n s of m o r e genera.1 f o r m , it is n e c e s s a r y to u s e g e n e r a l i z e d p o w e r s e r i e s a n a l o g o u s to t h o s e d e v e l o p e d b y V. I. 8 m i r n o v ([t0], C h a p t e r 5, S e c t i o n s 9 5 - 9 9 ) : f o r l i n e a r d i f f e r e n t i a l e q u a t i o n s . A l l t h e s e q u e s t i o n s , i n c l u d i n g the c a s e of e q u a t i o n s which c a n n o t b e s o l v e d f o r the h i g h e s t d e r i v a t i v e , a r e c o n s i d e r e d in [14-16]. 3. T h e m o s t e x t e n s i v e l y i n v e s t i g a t e d c l a s s of d i f f e r e n t i a l e q u a t i o n s is that of l i n e a r equation.s. T h e s e q u e s t i o n s have b e e n e x h a u s t i v e l y t r e a t e d in the m o n o g r a p h s of N. P . E r u g i n [2], L. S. F o n t r y a g t n [8], S. F . F e s h c h e n k o , N. I. S h k i l , , and L. D. N i k o l e n k o [12], and I. Z . S h t o k a l o [17, 18], w h i c h c o n t a i n a g r e a t n u m b e r of r e s u l t s a t t r i b u t a b l e to the a u t h o r s , and a l s o p r o v i d e d e t a i l e d b i b l i o g r a p h i e s . H o w e v e r , e v e n f o r l i n e a r e q u a t i o n s m a n y q u e s t i o n s s t i l l r e m a i n open and, to s o l v e s o m e of t h e m , the m e t h o d of p o w e r s e r i e s can be used. C o n s i d e r the v - t h o r d e r l i n e a r e q u a t i o n w~vl + r (z) w I~-1 + . . .
+ % (z) J + ~2 (z) ~ ' + % (z)~v + r (z) = O,
(37)
in w h i c h a l l the g i v e n ~0j(Z) a r e a n a l y t i c f u n c t i o n s of the c o m p l e x v a r i a b l e z = x + iy, s o t h a t e a c h of t h e m c a n b e p r e s e n t e d b y the c o r r e s p o n d i n g T a y l o r s e r i e s % (z) = ~
~
(z -- Zo)~,
i = 0, 1 . . . . . ~.
(38)
In p a r t i e u l a . r , ff one of the g i v e n f u n c t i o n s is a p o l y n o m i a l of d e g r e e m % (2:) = Pm (Z) = Po 2c Pl (Z -- Zo) -~- P2 (z -- Zo)2 -~, .. "7-'P,n ( Z - - ZO)rn,
(39)
t h e n its T a y l o r c o e f f i c i e n t s w i l l c o i n c i d e with the p o l y n o m i a l ' s c o e f f i c i e n t s , i , e . , in this c a s e Yio = Po; Yi! = Pl; " " " ;
~lm = Pro; Yin = 0
for
n ~ m + i.
(40)
I9t
We s e e k the solution to Eq. (37) in the f o r m of a s e r i e s oo
w = .~ a n (z - - Zo)~,
(41)
tt~0
p r e s e n t i n g all the d e r i v a t i v e s of w, in a c c o r d a n c e with the p r e v i o u s l y adopted notation of (10), by the series w~ = ~ %+~t,~t ,. __ zo)~,
(42)
n=O
where
a{u} ~ ~n ~ ~~ ~n ~ 2~
(n+•
u-----l, 2,
(43)
v
and, in p a r t i c u l a r , a n(it+ l
9
~
an+j
=(n+l)a+l;
~2~ _~ "*n+2
oo
an+2
=(n+l)(n+2)a.+2;
99
Substituting s e r i e s (41) and (42) in our o r i g i n a l equation (37), a f t e r the obvious t r a n s f o r m a t i o n s , we a r r i v e at the r e c u r s i o n f o r m u l a a'~+v =
Iv-ll + ' ' " +[Y3,~ 4+2 ] + [Y2nan+ll + [Ylnanl + Yon [Y'~.a.+~,--l] --(n + 1)(n + 2)...(n + v) '
(44)
which also s o l v e s the p r o b l e m we have b e e n p o s e d s i n c e , by its m e a n s , we can e x p r e s s all the s u c c e s s i v e c o e f f i c i e n t s , a v , av+l . . . . . of s e r i e s (41) in t e r m s of its f i r s t v c o e f f i c i e n t s a o, a I,
a 2,.-.,av_
I.
We u s e f o r m u l a (4) f o r the m u l t i p l i c a t i o n of s e r i e s in o r d e r to c o m p u t e the quantities [ylnan], [T2nan+l] . . . . . E x a m p l e 4. We shall c o n s t r u c t the g e n e r a l i n t e g r a l f o r the equation w" = - - :v cos z.
(45)
In the given e x a m p l e , v=2;
~o=%=0;
co
(~I = COSZ =
zL~V
Ylngn'J
~In---~Yn; Y2. = (-(2u)! '1)"" Y2.+l = 0;
n=0
s o that the r e c u r s i o n f o r m u l a of (41) takes the following f o r m : %+2=
[7.anl (n+ 1)(n+2)
; a o=w(z0);
a 1--w'(zo).
(46)
C o n s i d e r two b a s i c or f u n d a m e n t a l solutions, wt(z ) and w0(z), which we d e t e r m i n e f r o m the following initial data (where, by u s i n g the c h a n g e of v a r i a b l e s z* = z - z0, we can always s e t z 0 = 0): f o r the f i r s t f u n d a m e n t a l solution, w = wt(z), w(0) = + l;
w'(0) = 0 ;
(47)
w'(0)= +1.
(48)
f o r the s e c o n d f u n d a m e n t a l solution, w = w0(z), w(0)=0;
In this c a s e , the g e n e r a l solution of Eq. (45) will be w (z) ----Awl (z) + Bwo (z),
(49)
s i n c e w(0) = A, w'(0) = B. By s u b s t i t u t i n g the initial v a l u e s of (47) and (48) into the r e c u r s i o n f o r m u l a of (46), and u s i n g f o r m u l a (4) f o r the c o m p u t a t i o n of the quantities [Tna n] we obtain, as the solution being sought, the two s e r i e s , c o n v e r g e n t o v e r the e n t i r e (unclosed) c o m p l e x plane:
192
I
+2
i@yw,, Ii
~
\ -
\\
/
I 1
Fig. 2 z~ 2 z~ 9 z6 55 z 8 w z ( z ) = l - - - ~ - [ - t - - ~ -. - - ~ . +--~-.I z3
w~
4 za _
-1--~.
476 r~ --~-z +..4
19 z7 145z ~ ~ "+ 9[
1426z n 11I +""
(50) (5i)
F o r wt(z), t h e s e c o m p u t a t i o n s (with 7 - p l a c e a c c u r a c y ) , a r e p r o v i d e d in the f i r s t t h r e e c o l u m n s of T a b l e 5. E x a m p l e 5. W e now c o n s t r u c t the g e n e r a l i n t e g r a l f o r the e q u a t i o n (52)
co" = - - Z ~ . S i n c e the i d e n t i t y
gkco =. Zk E anZn . 2 an gn+k ~ E an--kgn' n~O n~O n~k is v a l i d f o r a n y i n t e g r a l ( p o s i t i v e o r n e g a t i v e ) k, we c a n then e a s i l y b r i n g the r e c u r s i o n f o r m u l a (with z 0 = 0) f o r Eq. (52), a f t e r u n c o m p l i c a t e d t r a n s f o r m a t i o n s , to the f o r m : d~ n
.
~
a'~+ 4 ---- (n + 3) (n + 4) '
a 0 = w (0);
~
a,
(0);
a 2 = a 3 = 0.
(53)
In this c a s e , f o r f u n d a m e n t a l s o l u t i o n s wl(z ) a n d w0(z), d e f i n e d f o r the s a m e i n i t i a l c o n d i t i o n s (47) a n d (48) as in E x a m p l e 4, we o b t a i n
Z4
Z8
wz(z) = I - - - - 3 - ~ +
3.4-7.8
g12 3.4-7.8-11-12 "+''";
(54)
wl ( - - z) = + w~ (z);
g5
w~
+
g9
ZI3
4.5-8.9
4.5.8.9.12.13
+'";
(55)
wo ( , - z) = - - w . (z).
For these series, mediately obtain
the r a d i u s of c o n v e r g e n c e is 12 = ~ s i n c e , f r o m r e c u r s i o n f o r m u l a (53), we i m -
R 4 = lim
a,~
= (n "-F 3) (n -b 4) = ~o.
n..~ an_t_4
193
y
§
I
-4
i
:/ I
Ill
l-e /
i
)
0
I
I
7
I
t
i i
1
-5 I
Fig. 3 D i f f e r e n t i a t i n g (54) and (55), we obtain the s e r i e s f o r the d e r i v a t i v e s
w; (z) =
z~
z'
z '1
3 -1- 3-4.7 Z4
3.4.7.8.11 + " ' ;
Z8
wo(z) = 1 ~ - +
w;(--z) = - - w~(z);
(56)
Wo(--z) = +w'o(z).
(57)
Z 12
4-5.8
4.5-8-9.12 W ' ' ' ;
We obtain the m o s t c o n v e n i e n t f o r m u l a s f o r c o m p u t i n g the v a l u e s of t h e s e functions b y u s i n g r e c u r s i o n f o r m u l a (53) and p r e s e n t i n g the s e r i e s in (54)-(57) in the f o r m : r
wx (z) =
An (z);
An = anz";
__ z4An . An+, = (n -{- 3) (n --{-4) '
n~O
A1 ~---A2 = Aa = 0;
A o = 1;
W[(Z) =
~
An (z);
/~n = anzn-:;
An+7= (n + 4) (n q- 7) i
n=0
A ~ = - - - ~z~- ;
Ao = 4 , = A ~ = o ; co
-
Wo (z)
=
A~ (z);
z%
.
An+4= (n --}-3) (n + 4) '
n=O
Ao=A;=As=0; 9.
~o(z) =
An(z);
A~=z;
-
A:,+,
zA;~
.
n (n + 3) '
n~0
A; =
A"2 =
Az "" = 0;
A~ = 1,
(58)
T h e z e r o s of functions wt(z ) and w,l(z ) a r e m o s t e a s i l y d e t e r m i n e d by N e w t o n ' s f o r m u l a , giving u s , f o r the r o o t of the equation wl(z ) = 0,
~i (zn) w h e r e z n is the n - t h a p p r o x i m a t i o n to the z e r o of function wl(z ).
194
(59)
A n a l o g o u s l y , f o r the equation w 1' (z) = 0 we have wl (z~) z~+~ = z~ + ?~v~ ( z ) since
(60)
'
w vT = - - Z 2 W .
The z e r o s of the functions w0(z) and w'0(z ) a r e d e t e r m i n e d a n a l o g o u s l y . T h e points of inflection of function w(z) c o i n c i d e with its z e r o s s i n c e , if W(Zn) = 0, to Eq. (52), we a l s o have that w"(zn) = 0. On Figo2 we have c o n s t r u c t e d the g r a p h s of the and w0(z), as well as t h o s e of t h e i r f i r s t d e r i v a t i v e s , wit(z) and w0'(z), f o r r e a l v a l u e s of z = x. By u s i n g the fact that functions wl(z ) and w0t(z ) a r e even, while functions w0(z ) and one can e a s i l y extend t h e s e g r a p h s to the n e g a t i v e s e m i a x i s .
then, a c c o r d i n g functions wi(z ) the a r g u m e n t , w l ' ( z ) a r e odd,
F o r l a r g e v a l u e s of Izl, s e r i e s (54)-(57) begin to c o n v e r g e m o r e s l o w l y and, in this c a s e , one can obtain b e t t e r r e s u l t s by m e a n s of analytic continuation. To this end, we p r e s e n t the o r i g i n a l equatio~ (52) in the f o r m (61)
w ~ = - - [(z - - Zo)~ + 2z o (z - - Zo) + z~l ~.
T h e s o l u t i o n to Eq. (61) will be the s e r i e s (62) rt~O
while the r e e u r s i o n f o r m u l a f o r d e t e r m i n i n g the new c o e f f i c i e n t s , a n , will now take the following f o r m :
a~+4 = - n>--2;
a2 + 2Zoan+, + z}a*.,~• (n § 3)(n + 4)
a'__~=a2,=O;
a;=w(z0);
(63) a~=~'(zo).
If we now d e t e r m i n e , with the s p e c i f i e d a c c u r a c y , the new initial v a l u e s a o = w(z0) and a~ = w~(z0) at points z = z0, u s i n g f o r m u l a s (54)-(57), we obtain the p o s s i b i l i t y of c a r r y i n g out f u r t h e r c o m p u t a t i o n s by m e a n s of the s e r g e s in (62). If z 0 is so c h o s e n that lz--zo[
< [zl,
then s e r i e s (62) will c o n v e r g e m o r e r a p i d l y at the given point z than will s e r i e s (54) and (55). We apply our r e s u l t s to the solution of the g e n e r a l R i c c a t i equation which, b y m e a n s of the s u b s t i t u tion g =
W~ -
-
-
(64)
-
c a n always be r e d u c e d to a s e c o n d - o r d e r l i n e a r equation. c o n s i d e r e d in E x a m p l e 1, to the equation
In p a r t i c u l a r ,
this s u b s t i t u t i o n b r i n g s Eq. (16),
~ " ~-~ ~ X ~ .
In this c a s e , a solution s a t i s f y i n g the initial condition y(O) = +1 will be //and
w~(z)--wo(z) ~ (z) - - wo (z) '
y(O)
~(z)
= wj (z)
9
0--1
= -{-, 1;
z =x,
(65)
1 -- 0
the z e r o s of the function --
~o
(z) = 0
(66)
will be the p o l e s of the R i c e a t i equation (16), and c a n be e a s i l y d e t e r m i n e d by N e w t o n ' s f o r m u l a . We find the initial a p p r o x i m a t i o n g r a p h i c a l l y ; on Fig. 2, ~he c i r c l e s m a r k the f i r s t t h r e e p o s i t i v e r o o t s of Eq. (66), while the c r o s s e s m a r k the two n e g a t i v e ones. In each s u c c e e d i n g step of N e w t o n ' s f o r m u l a f o r the given e x a m p l e we double the n u m b e r of c o r r e c t d e c i m a l p l a c e s , so that, by p e r f o r m i n g only t h r e e i t e r a t i o n s each, we d e t e r m i n e the f i r s t five p o l e s of I l i c c a t i e q u a t i o n (16), in which all the digits e x c e p t p e r h a p s the last are correct:
195
z=--3,3449074716591; z=--2,2233783825262113; z = +0,969810653931080907; z = +2,651806921116737922; z = +3.645776312779421937. T a b l e 5 g i v e s a l l the c o m p u t a t i o n s p e r f o r m e d in the s e c o n d s t e p b y f o r m u l a (58) f o r the d e t e r m i n a t i o n of the p o l e z = - - 2 . 2 2 3 . . . (the z e r o t h a p p r o x i m a t i o n f o r w h i c h was z 0 = - 2 . 2 4 ) , w h i l e F i g . 3 s h o w s the c o n s t r u c t e d s o l u t i o n f o r the R i e e a t i e q u a t i o n y' = y~ + x2;
g (0) = + 1.
T h e r e s u l t s p r e s e n t e d a b o v e c a n , in an o b v i o u s w a y , b e g e n e r a l i z e d to s y s t e m s of l i n e a r e q u a t i o n s of the t y p e of (37), b u t s p a c e l i m i t a t i o n s p r e v e n t u s f r o m e n t e r i n g into this m a t t e r in this p a p e r . 4. W e t u r n now, v e r y b r i e f l y , to the q u e s t i o n of g u a r a n t e e i n g a g i v e n d e g r e e of a c c u r a c y of t h e c o m putations. An e x i s t e n c e t h e o r e m a l l o w s us to p r o v e that, f o r a d e f i n i t e c l a s s of d i f f e r e n t i a l e q u a t i o n s , one can p r o v i d e any p r e v i o u s l y s p e c i f i e d d e g r e e of a c c u r a c y . I n d e e d , a c c o r d i n g to this t h e o r e m , f o r the e q u a t i o n dw d z -- f(z; w),
(67)
w h e r e f(z; w) is an a n a l y t i c f u n c t i o n of two c o m p l e x v a r i a b l e s in r e g i o n D
Iz--zol
l ~ - - ~ o l
b o u n d e d in D, If(z; w)[ ~ M a n d c o n t i n u o u s , when z and w a r e on the b o u n d a r y c u r v e s t h e m s e l v e s , a b l e b y the p o w e r s e r i e s w =c o +cl(Z--Zo) +...
+ c n ( z _Zo),~ + . . . ;
there exists a solution, representco =w(z0),
(68)
w h i c h c o n v e r g e s at l e a s t in the c i r c l e ]z--z0l
Q=min
a; ~ - ;
Q4R,
(69)
w h e r e IR is the t r u e , g e n e r a l l y unknown i n i t i a l l y , v a l u e of the r a d i u s of c o n v e r g e n c e of s e r i e s (68). In s u c h a c a s e , if t h e r a d i u s of g u a r a n t e e d c o n v e r g e n c e is
t h e n , b y t a k i n g a s u f f i c i e n t n u m b e r of t e r m s of s e r i e s (68), we c a n p r o v i d e the r e q u i r e d a c c u r a c y f o r a l l f o r lz - z0l < a o f i n t e r e s t to u s . If, now, p < a , we then have to u t i l i z e a n a l y t i c c o n t i n u a t i o n , c h o o s i n g a new c e n t e r of the s e r i e s in a c c o r d a n c e with the c o n d i t i o n
1
tAz0I ~ ~-~,
(70)
a f t e r w h i c h t h e e n t i r e p r o c e s s c a n b e r e p e a t e d . As the r e s u l t , we w i l l e i t h e r have w i d e n e d the d o m a i n of h o l o m o r p h i c t t y up to the r e q u i s i t e d i m e n s i o n s o r we w i l l have g o t t e n a s c l o s e a s d e s i r e d to the s i n g u l a r p o i n t c l o s e s t to the i n i t i a l z 0 of Eq. (67), w h i c h a l s o d e f i n e s the t r u e r a d i u s of c o n v e r g e n c e R, of s e r i e s (68). It is n e c e s s a r y to p e r f o r m a n u m e r i c a l c h e c k on the s e l e c t e d n u m b e r of t e r m s , N, of the s e r i e s b y s u b s t i t u t i n g a p o l y n o m i a l of d e g r e e N, w h i c h we in f a c t r e p l a c e b y s e r i e s (68), in the i n i t i a l e q u a t i o n , a f t e r w h i c h we v e r i f y d i r e c t l y w h e t h e r this p o l y n o m i a l s a t i s f i e s Eq. (67) w i t h t h e r e q u i r e d d e g r e e of a c c u r a c y . In s o l v i n g c o n c r e t e p r o b l e m s , it is a l s o n e c e s s a r y to v e r i f y t h a t f u n c t i o n f(z;w) s a t i s f i e s a l l the c o n d i t i o n s g u a r a n t e e i n g u n i q u e n e s s of the s o l u t i o n . A w a r n i n g in this r e s p e c t is f u r n i s h e d b y the e x a m p l e ,
196
constructed by M. A. Lavrent'ev [4], in which although function f(z;w) of Eq. (67) is continuous, nonetheless not one, but at least two, integral curves pass through any point in region D in any neighborhood of it. The results initially obtained for Eq. (67) were thereafter generalized to Mgh-order systems of such equations, as elucidated in detail in [i, 3, 5-7, 9, 11]. LITERATURE i. 2. 3. 4. 5. 6. 7. 8. 9. I0. ii. 12. 13. 14.
15. 16. 17. 18.
equations and to
CITED
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t97
