OA 431 B3 UC-NRLF EXAMPLES ^Dlutioni^ FUNCTIONAL EQUATIONS CHARLES BABBAGE, a.m. f.r.s. l.&e. fx.p.s. AND SECRETARY TO THE ASTRONOMICAL SOCIETV OF LONDON. • '•'•• m m NOTICE. The object of the following Examples of Func- tional Equations^ is to render a subject of considerable interest, more accessible to mathematical students, than it has hitherto been. It is, perhaps, that subject of all others, which most requires the assistance of particular instances, in order fully to comprehend the meaning of its symbols, which are of the most extreme generality ; that assistance is also more particularly required in this branch of science, in consequence of its never yet having found its way into an Elementary Treatise. Oct. 20. 1820. 437522 OF FUNCTIONAL EQUATIONS If a function a is of such a form, that, when it is twice performed on a quantity, the result is the quantity itself, or if a^ (x) = Xy then it is called a periodic function of the second order, if a" (x) = Xy then it is termed a periodic func- tion of the n^^ order, thus when a(x) = a -^ x the second function, or a{ax)=:a{a — x) = a — (a—x)=za — a + x=:x. If a(x) =— L_, then a^X=:a{ax) = I == ^ ~ "^ = ^ " ^ , 1 1 — A? — 1 X I - X and - » ax — I 1 a' a: = a* a a: = = _ a. X = 1 — I — X = Xy the first of these examples is a periodic function of the second, the last is a periodic function of the third order. Prob. 1. To find periodic functions of the second order. Since such functions must satisfy the equation \lf''x = Xy we have or yj/ must be such a function, that it shall be the same as its inverse ; if therefore y :=. \'/ x, we have also .r = \// — 'i/ = \//^, t A (2) or if X and y are connected by some equation, it must be symmetrical relative to x and y, y or "^ x must then be determined from the equation =^ jP { 7, ^ j = 0, for instance, if x -h ^x — a =: 0, \l^ x s= a — x, or if X \l^ X = a"^, \f^ x = -^ , X Another method of determining such functions is as follows : since v/^ J? is of such a form that ^^x = x any sym- metrical function of x and \/^ x remain constant when x is changed into yjy x thus i^ { JT, \l^x \ becomes F { \{^x, \/^'' a } = F { y^ x, x i y if therefore, we can find any particular solution of the equa- tion \lr^ X == Xf containing an arbitrary constant we may sub- stitute such a function for it, but yl^x=a—x is a particular solution therefore \!y X = F{Xy xf/^ x) — X, or x •{■ y\r X =^ F {Xy y\f x)j and by changing the arbitrary function into another of the same form, we find F \ Icy ^ \ = 0, as before. These two methods of determining periodic functions of the second order, are not so convenient as a third process which can be extended to all orders. * Bars placed above quantities under the functional sign, in- dicate that the function is symmetrical relative to those quantities. (3) Assume \!/ x =:
.r =x = X, which will be verified if f(v) is a particular solution of J^ v=.Vy and if ~ * is such an inverse value that "~ *(/> v = i;> hence the solution of the equation is yjy X =z (p~^f(pX, one solution is and hence ^x z=. (h — ^ ( J more particular cases are y^X = y^ X =. a — X \ — 'd X a" , \/a ar^ - a" \I/^X== -— \jyX=: a c — c^ X ' X , ax — d^ . 1 \lrX=: -4^ X =: 1 - X ^"^^ (^zr^y >/^^=-iog(i-.aO (a a?" — a^) ^ yf^x = V _>» y}.X=:log(as'=-a'')-x X I a + b X , , , Prob. 3. To find periodic functions of the /z'^ order, or to solve the equation v/^" x =z x. Assume as before ^x =^ cp-'ftpx then it becomes (5) which is verified if / is a particular solution of/' x = x, an4 if ) ~ ' is such an inverse function that il)~' (px = x. It now remains to find particular solutions of 4^"" x = .r which may be accomplished in the following manner : let fx represent - — — then the «"' function will be of the same c -{-dx form, or C„ + DnX where ^„, B„, C„, D„, are functions of ^, &, r, ^, and «, these may be so determined that Z)a=0, ^„ = and B^ = Cn all which conditions are satisfied, if ^^ — 2 & ^ COS d = - ( 2 + 2 COS n ^ hence (l)X = (p =.-^ a + b^x Q Jz TT b"" ~-Q,b c cos + c'^ n c — ( 2 + 2 cos ^h-K — )a n / (p j! a more detailed account of this method of solution may be found in a paper by Mr. Horner in the Annals of Philosophy, Nov. 1817. Instances oi \fy*x = x are / 1 1 2 1—^ 2 \l^x = 2 — X X — 1 X \lr X = \lr X = xjy X == 1 •^x 1 — X 2 ac - c^x a -\- b X c b' + c" X. 2« (6) (2 a:** - Qy \// J7 = log 2 — .T + log (e' — 1). All those cases which satisfy the equation xj^^ x = Xj also fulfil that of \|a* .r = x, as well as all those which fulfil any of these equations \/^' j: = — :r, -v//^ ^r = - , or more generally X x|/' x—a Xy where « j; is a particular solution of the equation \lr^ X = X. The following particular cases satisfy the equation \jr^X = X, 1 , Sx — 1 ^ 3{l - X) ^ Sx 3 , ^-2 y\fX = y]/ X = 3 — X Sac — c oar — l , 3 + 3 X xlr X = S S^ X = ^ X 3 — cr a + bx 4rX = be + c'' ■X 3a ^^ = K^"--3) \/^cr = log 3 — ^ + log (e' — 1). The principle on which the solution of the functional equation F { x, ^ Xy \l/ a x \ =0 depends, where a'x^zx^ is that by substituting ax for .r we have another equation F \axj Kjy a Xy xj^ X \ = 0, between which and the given equation we may eliminate yp a x and the result will be the value of v/^jr a few examples will illustrate this method. (V (1). Given v/. (a ) + «>/^(- a) = :c« by putting — x for a^ this becomes and eliminating \/^ ( — x), we have \// j: — a' \/^ ^ = *^ — fl (— xT, hence ^ 1 -a* (2). Given yj^ x — a^r^ =:e* 11 ~ put - for :r, "v// a^l^x == e X and yjy'X = (3). Given (v// :r)^ . v/. 1 - a" 1 — T 1 + X 1 — 2^ put for Xy it becomes ^ 1 4- a: yy^ I + x/ 1 + ^ eliminating x/^ ^ ~~ ^ by means of the former, we find (8) (4). Given xj^ x -^ —^ xj^ ^/O - x") = I + x' putting V(l — ^t'') for x, we have and substituting this value of \/^ ^1 —x"" in the former equa- tion hence and yjy X zi x^, (5). Given ^^ +^ ^^"^> =i 1 +X/.X 1 + .^(-x) put \/^i a^ == • — thus the equation becomes ^lr^x + X yp^i (^— x) = 1, and changing x into — a; we have 4^1 { — X) — ar\/^, (a) = 1, by which eliminating \l^{ — x) from the former, we find ^' 1 + X^' hence yjyyX 1 — X ylr X = I — \l^^x X + x"" (6). Given yj^ x + ^ "^ ^ ^' = c. putting - for x this becomes (9) vf/ _ + (1 -^ X) \I/X ss X r and by eliminating \/r _, we have , 1 (7). Given x/^ i -f- j: v/^ ( 1 - i^^) = 1, putting ]—x for .r, we have v|r(l ~ i) + (1 _ jr)>^(i') = 1, whence, by elimination, 1 - i 1 - r -vl^ r = 1 -^ j^(l -. .r) 1 -.r ^- x' (8). Given /" + .__A(L:l£) = i, put v/., X = -Jl^, then will v/^,(l -ar)= tlLjUf), ^ , and the equation becomes \/^, j: + .r v|/, (1 - x) ss 1, the same as in the last example; let /.? represent the solution there found, then whence -^ >l^ X -- X , l/ X / X — 1 1 — a if we take for fr its value ■ , we have w r = ; . X f B (10) Incase the equation is symmetrical with regard to \/x>r and Vr a ,r, the process of elimination apparently becomes illusory. By a peculiar artifice this difficulty may be over- come, and it happens rather singularly that in all these cases, the solution which is so obtained contains an arbitrary func- tion, and in general the solution is the most extensive which the question admits of. (9). Given \j^ .r = 4^- . X If we put - for .1 , this is changed into ^ ^ = \j^a;, the same X X as the given equation ; it is therefore impossible to eliminate. Let us now suppose yj^ x = a 4^ — \- h, which becomes the given equation when a = l and ^ = 0. By putting _. for x this is changed into ■KJy - = n v/x X 4- h, X and eliminating \// -, we have , __ ah ^- b _ _^_ if 6 = and r/ = 1, this becomes a vanishing fraction whose value is any constant quantity c, and we have \//j: = r, which fulfils the equation. This is a very limited solution, but the following plan will lead us to much more general ones. Take the equation . 1 \^ X =. a ^/ ~ -\- "J (p Xy X which coincides with the given one when ^' = 0and<7= 1 ; also (p X is any arbitrary function of a ; putting - for :( , we have (M) and by elimination, t a (b- + (j) X Let « become 1 + and v become at the same time, then 1] 1-1.1+2.0 + 0^) -2.0 + 0* -2+0 2 and the solution becomes 1 Vx X ~ — , or changing the arbitrary function \lrX=(})X-\-
is indefinite. This solution is, in fact, nothing more than an arbitrary symmetrical function of x and - , and may be expressed thus ^^ = xO. 0- Precisely the same course of reasoning will produce tlie solutions of the following equation. (10). yly{x)z= yi^{a - x) V^ X =: xix, a - x). (13) (14). \/^ .X = \^ (a a), \^ X = ;^ (JT, a x), whcrc a* ^ = 1\ (15). The objection which has just been stated occurs in the equation \/^ x -f ^ ( ) = ^, and a similar mode of proceeding will obviate it. The givers equation is a particular case of f + a \lr ( — t — y = c -h v^ Xt with which it coincides, if o = 1 and ij = ; putting for X in tliis, we have and elimination produces If v = and « = 1, this gives \^ .1 = - -h ) t — (rl-i) in which the function > has been changed into another simi- lar one. 16- Given v/^(l -i- *) + \'^ (I — = 1 - i% put 1— i for i, then (13) x]!. X + x/> (2 - a:) = 1 - (i- - 1)» = 2 Jf - .r% this is a particular case of the equation \l^ X + a \l^ {'2, — x) = 2 .1 — .1* + V ipx, with which it agrees li v—0 and fl=:l ; changing x into 2 — r and eliminating xjy {2 — z) from the result, we find x/..r e= \ + { we have
(b X — a (t) i - — X ) V
-J/ a: =
1 + a 1 - «^
making /?= 1 and v = 0, and changing (p x in 2 r, we have
^,.r = i -^.r + 0(^-^)
and therefore
In case the coefficient of yf^ a x in the equation
\p' X + fx 4r ax =fx, IS of such 2. form that fx .fa .r = 1 ,
the denominator will vanish, and wo must then have recourse
to an artifice similar to that which has already been ex-
plained.
(20). Ex. 1. Let v|, r + jr"" x/^ i = x''.
X
put v(^ X + (X^'' -^V(Px)^|^l= r"
(16)
which coincides with the given equation if vnOi then chang-
ing r into -J we have
^
I -\' (x-"" + V(f>i^ ^^X- I -«
X
and by elimination
\'/ X zz
.t~"03:
:r»« i + x-^"
\x — \^ X — 1
which is a particular case of
Vx — 1/ X — 1
with which it agrees, when a = — I andi;=:0.
Put^ for r, then
X — 1
whence by elimination,
(18)
(ct — i)-^- ] I + a + v(px
XJy^ X ZZ - * '■ ■ ' ■ '
1 — (a -^ v x
" x-\ ' /^ ^ \*
and restoring the value of \/^, x, we find
"^ X =
\r^ — 2 X /', ^= ^ — ^^ ; ^ 0^__ <^^(1 _ ^^2^^
hence
2 v/(l - ^')
(19)
It would not be difficult to shew the identity of these
two apparently different solutions.
(23). Given the equation
\l^x . V^(l — '^) _ ,
T" — tt: — ^^ ^>
put V^,a- = • .n ; Ti ^l> then x^. ( 1 - :r = X-L-_-/ ;
and the equation becomes
^1^^' + ^.(1 —x) = 1,
whose general solution is
\/^i jc = ^- — ;
0-r+0(l-x)'
hence .
putting 1 — .r for x, and eliminating v/^ (1 — x), we find
[0(1-^) + «^^r
(24). Given (^|. xf + (^ -)' = ^L±_f! v^ x . v/. f! •,
^ X ^ X^ X
a"
divide by v/^ a . \^ — then
^x f_ _ X* + a*
~~aF ^l^ X '^
putting \'/i .r — ^ 3 this becomes
(20)
^l^ + ^^ — = — - — >
a particular solution of which is ^/^l ^ = x*; hence
= .1"^ or v/ X =L ^^ -v// —
, « X
X
and the general solution of this is
\^-x^\
(25). Given riL±f + x ^ = 1 -f ^%
v/^ X -i- .2?
put \^, x = — , then the equation becomes
X
^, X + x'' x/., i = 1 + a:%
whose solution is >i/^x •=. ^ — , ^-I— - ;
X
, a? + ^x ix'' + 1)0. r
hence ^ = ^^— ;
putting - for x and eliminating -v/^ - , we find
.^x= ; i — ; — '—rr^^^ I +'"*")
X ^ 1/
(21)
(26). Given (v/. xf' . {^|r - xT - (^ ^T • (^ - ^)'" = ^ .r ;
putting >//, X = (^^ j:)'^ . (Vx — xf, it becomes
^/-.^ - ^^.(- ^)= '^^>
whose solution is ^t v = x + x (^> ~ '^)>
hence
and by the process for eliminating yjy {— x), we shall find
I \ x{^» '^T)-^r ] '^J
(27). Given y^rx ■{- fx ,^ o.x = /» x, where ax \^ such
a function of x that a'' a = cT ; putting « x for i', we have
■y\r _iL_|
(35), Given ^x ^ a^(^ n/Cj:" - 1)\ _ ^,„^
put — for X, it becomes
^ X
^V .r y Vn/(1 - x')/ x'^
Again, put V_l for i, and we have
A, f . — 1 — a-ly X :iz «
by eliminating ^ ^\/('^' ~ ^^^and ^^ (^~ ~^) ^^"^
these three equations, we shall find
(36). Given +. + + (iJ:|-) + +4^) -
the function ;;— is periodic of the third order^ and by
1 ■"" *3 X
the process of elimination
/ \ + X \
^.x =
(37). Given -i~ + ^ + ■- = «•
Putting i]/, ;c = — , we have
V -^x
Tl «-
X I — X
whose solution is
n A\ ^-^ t A^ 'K' . ... rf>
^.^
1 ^^-l'
I — X X
hence
Y, +
whose complete solution is
(pT + (pax fx
tan~' ^ X = tan~' ,
hence
\^ i- = tan < 1 tan ' >
C (/> 2^ + a x r .r >
(p X -i- (p aX f i
this process is analogous to one employed by M. Laplace, for
the integration of a similar equation of differences.
* Journal de I'Ecole FolytccnUjiLc , Cuh. l<5.
(21)
(44). Given
,,,H..(iii)..(-l)..,t^).,
1 +.1-
1 -X
being a periodic function of the 4'*" order.
^x= *^
.1-1 V 1 — a: X X -{- 1/
j:.(i-x)'^(. ' T:::ri* x 'x + ii*
(45), Given ^.{x, y) + v^ (^^\ i^ = l,
Vx 1/ X
(♦6).
<^(*, y) + ?j)'fMy «^ v, A:^2/),
(28)
where u^x = a; ft"^ y = y and f{^x^ y) is such a function that
/(^, ?/)./(« .r, /3 2/)= 1,
then v/^Cr, v) = n^/(«^> /^.V) /x(^, «^, .VW^?/)
/(.r, y) /, (^, x)
^ (^^ 2/) = X (^ + ^, 2 y),
(53). Given ^/. (.r, y) = (£)' >/. (^, ^)
(54). Given x/. (tt - .r) = ^-^,
differentiating f yf, (n - x) = ^1^,
(2!>)
putting TT — .r for x In the given equation ■
ax
1 i« • • d \U (tt — jc) ,
and eliminating — i— ^ , we have
whence by integration
y^ X =z b cos J? 4- ^ sin T,
and it will be found that c = — ^ ; hence
y^r X =. b (cos X — sin X).
(55). Given i. (*, y) = d^{x,a-y) ^
d X
put a '- y for y^ then
differentiate this relative to a, then
r/ ^{x, a - y) _ ^^ v/r Tj:, y)
which being substituted in the given equation produces
whose solution js
y^{Xy y) =: e^0?/ + E-^0,^,
*/) and 0, being two arbitrary functions so constituted as to
fulfil the given equation, in order to determine them, put
48—1/ for y and differentiate relative to x, then
d \l^ {x, a — jj) , , ^ , , ^ .
(30)
hence
tpi/ = (^^J + (p, {xy) :
to determine the form of (/> and ^j, we have
d^(x')
= i^^^) + xy '(xy)=--l-^ ' ( J-)
The first of these multiplied by d{xy) may be put under the
form
whose integral is
xy
ix,y)=^xj,.xQ^l) +
y x>
fd{xy). f— )'x. i^yy — )
\j.'i// V xy/
(63). Given i±j£iJL}l} rz "LtMj^ ,
dx dy
where a^y =^ y and /3* x = x a process nearly similar to that
(36)
by which the two last equations were solved will lead to the
partial differential equation
d a y d" \l^ ( JT, t/) __ fLUjE. d' \lr (Xy y)
dy d x"" d X dy^
(64). Given y\ra.X — y\ry\rX — y\/''X,
It is evident, that whatever be the form of a, this equation
can always be satisfied by assuming \|/- x = a ^r, hence the
solutions of the following equations,
, ax
>\rX = -—
o -\- C X
(65). Given ^ (2a - j:) = x/.» x.
Put y^yX = (p—'fipX,
then \lr'^X=:(}> — ^f(p