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 =: <p — '/(pr, then 
 
 V/^ :« </)-700-'/</>.r = <p-T<}>x, 
 this must be equal to x or 
 
 this equation will be fulfilled if f'*v = v, or if /is a particular 
 solution, and if also <^ "" ' is such an inverse function that 
 <^ -~ ' <^ V s=v. If therefore is arbitrary, and /is a particular 
 solution of y cT = X, then the solution of \|^* j: = a: is 
 
 \jy X = (p"^ 'y a:. 
 
 Ex. Let/j: = -, then V^a = 0--'(— ), 
 a: \(px/ 
 
 if /(Ar)=_-- , x^* = 0-»l-_ r_l; 
 
 from these may easily be derived the following periodic 
 functions of the second ord&r, 
 
 
 ^Iv ~ ^ 
 
 
 yy X — • ' 
 
 X — I 
 
 X — 1 
 
 
 1 — X 
 
 
 \j^ X SS 
 
 1 + X 
 
 v/. a; = \/l - o:^ 
 
 X + 1 
 
 , X 
 
 v'*' — I 
 
 >^or=tan-(^'"(''-^>) ^x=log(«-0 
 Vcos «. cos j:/ o \ / 
 
 \l^x =1 (a" - Af")'' yj^x =:x - log («^ - 1) 
 
 X 
 
 ^j^x= Y ^x =:tdn-' (a- t^nx) 
 
(4) 
 
 Prob. 2. Required periodic functions of the third 
 order, or such as fulfil the equation \lr^ x = x. 
 
 Assume "^l^x = <}>~^ f <P ^t then the equation becomes 
 
 \}y^X = (p~ ^f(p (p ~ ' f(p (jy — '/ipx = <p~^f^<l>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-\-<p~, 
 X 
 
 in which </> 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= \ + { <t>:t-a<p{2-x) \ .. 
 
 I — a' I — a* 
 
 or 
 
 x/. J = ^_f 1* 4- i ^:r - r/</>(2 - i) \ "" 
 
 i + rt ^ ^ ' 1 - «• • 
 
 If a = l and v = Of we have 
 
 2 t — a^^ 
 
 (17.) Given -— + 7 = 2, 
 
 \ + x\l^- 
 
 X 
 
 put —- = v/^, r then ^ = v^, - = , 
 
 ^ X -^ \^x 1 . / 1 I' . .1 
 
 ^ -+>/^- 1-1-2 x//i 
 
 and the equation becomes 
 
 x/^. .r + x/^, - = 2, 
 
 a: 
 
 whose solution may be found by the method just explained 
 to be 
 
 \/^. 2 = J -f .1 _ ^ ^ , 
 
 hence 
 
 , 1 
 
 1 + 0.t - ^i 
 
 X 
 
(14) 
 
 (18). Required the equation of that class of curves 
 which possess the following property, (Part IV. Fig. 1.) a 
 given abscissa A B — a being taken, then the product of any 
 two ordinates at equal distances from B^ shall always be 
 equal to the square of the abscissa a. If y = V^ i represent 
 the equation of the curve, then the condition expressed analy- 
 tically is 
 
 \ly(a — .?;) . -v// (fl -\- x) = d\ 
 Putting a — X ior i, and then log \'r (x) = \/^i x, we have 
 
 \l^, 1 -f- "^1 (2 ^ — x) = 2 log a, 
 whose solution is 
 
 yfy-t X = log a -^ (j) X — <p (2 a — X), 
 hence 
 
 logyj/ X = loga + (px — (p(^a — x), 
 and 
 
 ioga e.v — »(2e — .r) ^ , 
 
 yjyX = E X 6 X e 
 
 e 
 
 and changing the arbitrary function into log (p, 
 
 , (p X 
 
 (p(2a- x) 
 and the class of curves are comprehended in the equation 
 
 y = 1 , 
 
 ■ (p(2a — X) 
 (J 9). Given the equation 
 
 This is the equation on which the composition of forces is 
 made to depend in the Mecanique Cocleste, p. .5. 
 
(15) 
 Put \|y^ r for {xjy vY then it becomes 
 
 which is a particular case of 
 
 yj^.x + a V^,(^ — r^ =: 1 + v <p a. 
 
 Substituting - - x for x, this gives 
 
 and eliminating ^i ( ^' ) > 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-^"<pa -{- v<px .<p~. 
 
 X X 
 
 and when v vanishes, 
 
 xU X zz Z-^— — , (a). 
 
 X 
 
 The equation in this example may be solved difFerentlyj 
 as follows. Multiply by x-"" and it becomes 
 
 X 
 
 put Vx, 1 rz :r — " \|/ X, then 
 
 .r 
 
 whose general solution found by a process already explained 
 is 
 
 4r^x ■=. <t> X -\- (f>~ f 
 
 2 X 
 
 hence 
 
 r" 1 
 
 A X 
 
This solution differs in form from that which wa$ pre- 
 viously found, but it may be proved to be the same by the 
 following substitution ; since (p is quite an arbitrary function 
 
 this gives x^''<p^ + x — ^'cp x =: 2 and {a) becomes 
 
 x'' 1 
 
 2 X 
 
 exactly as the last solution. 
 
 (21). Given 
 
 
 _(x_~jiT-i_ 
 
 X - 1 
 
 - 1 
 
 put \/^, X = — — — — , then it becomes 
 
 \l/ X — 1 
 
 , / :r \ (x-lY~l 
 
 ^i^ - ^» ( ) = ' ;, > 
 
 \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<p x) ( a + V (p ) 
 
 and when ^ = — ] , and v =0, this becomes 
 
 . ^' -2x <t>x 
 
 " x-\ ' /^ ^ \* 
 
 and restoring the value of \/^, x, we find 
 
 "^ X = 
 
 \r^ — 2 X <p X ^ IT 
 
 .r - 1 
 
 ^^■^■Kr^) ' 
 
 
 X'' (p vCi - ^') + (I — ^'^)^ .1 
 
 this solution was found by pursuing the course so frequently 
 pointed out : another but not a more general one may be 
 obtained as follows : multiply by ^{ 1 — x"") ; then 
 
 V(l - x") xf^x + xyj^ V(l _ A^^) = .r V (1 - x^\ 
 putting v/(l ~ x'^) 4" X = yjy ii, we have 
 
 ^,-^ + ^. V(l - ^^) = x^il - x% 
 whose solution is >/', ^= ^ — ^^ ; ^ 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<xX +fa X .^ X =.f^ a ,r, 
 
 and by eliminating 4^ ax 
 
 If /j? ./a a: =: 1 and /i a; — /^ ,f^a x zz 0, then the solution 
 becomes a vanishing fraction ; also the general value of f^ x 
 is in that case /, x = \/if x) ,f.^ (x, a x) and the equation 
 becomes 
 
 \lr X + fx ,\l^aX — y/fx ./a (^, a x) ; 
 
 dividing this by \/(fx) and putting instead of — -r- its value 
 
 >s/fx 
 
 ^{fo.x) derived from the equation /a: ,fax-=:\y we have 
 
 \/{fa x) . Vx .r + x/(/^) . >/^ « X =/, (.r, a ,r), 
 which is a symmetrical equation, whose general solution is 
 
 ,, r \ 1 fti'V, ax) .<p X 
 
(22) 
 
 hence 
 
 <p X + a .r 
 
 (28). Given a + l^^x = -^{a + b x), 
 
 \|/ :c = ^ _ + t)^ X. 
 
 1 — b 
 
 (29). Given --^i:^ = 4, ( "^ ) 
 
 
 (30). Given _±£_ = + (_^) 5 
 
 4^ 
 
 ^-=^^^'^^-(-1)" 
 
 where n is arbitrary; it may therefore be changed into any 
 symmetrical function of x and . 
 
 (31). The three last examples are particular cases of the 
 equation 
 
 a -^^ X = '^ a Xy 
 
 whose general solution is -^ x = a" jr. 
 
 (32). Given ^ ^^y ^^ \ = ^ VTT2"J ' 
 
 1 + « .r 
 
(23) 
 
 (33). . Given a^aX = ■^a'' X 
 -^X = cl"" X. 
 
 (34.). Given 4/ a -f « 4/ ( ) = - • 
 
 is a periodic function of the third order, or of the 
 
 form a^ x = x; putting for x, we have 
 
 and in this again putting for Xy we find 
 
 1 — X 
 
 ^ ( ^ ~ ) -\- ay{^x= — i_ ; 
 \ X ^ X — ] 
 
 •^ ( ) and -vj/ ( J being eliminated between these 
 
 three equations, we have 
 
 ^x =— i— V-- a{l -X) +>_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, + <p^_^ + i, ^ 1 
 
 4/07 = 
 
 
(25) 
 
 Y:]8). Given \'/ a . V^ ■ . ^ ^-^ — = c^- 
 
 Putting ^|/^ X = log 4r X, wc find 
 
 ^,3: 4- y^, + v/^. — = log {c^)y 
 
 1 — X X 
 
 whose solution is found in the last problem. Changing (/j, 
 into log <pi, we have 
 
 Similarly if « :r be any periodic equation of the third order. 
 
 (39). \lrx + \l^ax + \l^a''x = at 
 has for its solution 
 
 (p X + (J) ax + (pa^ X 
 (40). yj/- X , xj^ a X , \lr a"" X '— C^, 
 
 has for its solution 
 
 (pax t(px+(paX+ipa'^Xl 
 
 (41). Given \1^ x + f x ,^a x =/, Xy where a^ x — x. 
 Putting successively a x and a^ x for x, we have 
 
 y^ a X ■\- fa X .^ a'^ X ^=^ f^a X, 
 y^a^ X + fo.^ X ^ X =/, a^Xy 
 
 and eliminating ^ax and v/^ «'* x from these three equations, 
 we have 
 
 1 + fx .fa X .y^«' X 
 t D 
 
(26) 
 
 (42). Given xj^ x + fi,\l/-a x =/, x where a" .r = x, 
 * a similar process of elimination will produce 
 
 ^. ,- - /i ^-/^^ .f,aJC+.,..fx ./a X ./g'— ^ 1 ./ a"-'l- 
 "^^ l-{-'\ffx.fax..fa''-'x 
 
 (4-3). Given the equation 
 
 14. yj? . (v/^ X + \/^ a .r) — v// x . \|/ a :c = 0, 
 
 where « a; is a periodic function of the second order, and/ J? 
 is any function symmetrical relative to x and a x 
 
 \lr X . xfy a X — 1 
 
 /^ = 
 
 yp- X + \l/^ ct X 
 
 consider v/^ x and 4" ax as two variables, and differentiate 
 with respect to them, then 
 
 d\ly X d4^aoc _ 
 
 "T" , . , n — ^-^j 
 
 1 + {^OCf (1 + x/^aj)^ 
 
 and by integration, 
 
 tan — ' v/^ .r + tan ~ ' \/^ « X = C = - -^r- > 
 
 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) + <p(-, -y) 
 
 <47). Given ^. (x, t/) + x^ v^ ^1' , --^ ^ = _£^ 
 
 Vx 2/^-1/ 5r-l 
 
 v^(x, 7/) ^ (y-J)-'^y0Cr, y) 
 (48). Given x^ (x, t/) +/(x, y)^{ax, ft y) == 
 
 ' 1 
 
 ^A^> ?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)<p{ax, (iy) +f{ax, fty)(p{x, y) 
 
 (49). Given ^l.{x, y) = ^}. (- ^, £ ^) 
 
 <50). Given xj. (,r, 2/) = v/. (Ahj) , ^Jfli^N ^ 
 
 ^2/ X / 
 
 (51). Given ^^ (.r, i/) = ^/^ (| y/l. , ^J7y) , 
 
 (52). Given yJ. (,i, !/)=>/, (^, 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/ = <p (a - y) and <p,y =:- <p, {a - ij\ 
 whose solutions are 
 
 hence the general solution of the equation is 
 
 A similar mode of solution is applicable to the three follow- 
 ing equations. 
 
 (56). Given v/^ {x, i/) = -j- x/^ (x, -\ 
 
 ax V y/ 
 
 \ y^ 7/ v 1/ 
 
 (57). Given ^{x, y) = J~ V- (^r, 7-^1-) 
 
 (08). Given ^r (.t, j/) = — v/, (a, a 1/), where a' y = y 
 
 (59). Given vKx, y) = -It^i-^ 
 
 where a is such a function that u^y = j/. 
 
 Substituting successively ay, a'^y, a^y for ?/, we have 
 
4x 
 
 If IN d^i^i y) 
 a X 
 
 From the given equation ^^ * — ^ may be eliminated 
 
 by means of the second, and from the result — -^ — ^ ^ 
 
 ax 
 
 may be eliminated by the third equation, and continuing this, 
 we should find 
 
 V- (^. 2/) = — ^^ . 
 
 the solution of this equation is 
 
 "4^ (Xj y) == i' ^y -}- B -- ^ ^, y + sin 2^ . </)o 1/ + COS X . 0., y, 
 
 *Pi </>!> 02> 03 must be determined so as to satisfy the given 
 equation, taking the differential and putting a y for i/, we 
 have 
 
 U M/ ( X CI lj\ • 
 
 * ^ =i^<puy^i—^ tp^ a l/-f-COS X .<p, ay — SmX .(p^aJ/ 
 
 ax 
 
 the first condition to satisfy is 
 
 02/ = 0«2/> 
 which gives 
 
 <py = x(^' "y» «'i/> ^'.v)j 
 
 the next condition is 
 
 <Piy =- 0. «2/- 
 whose solution is 
 
 0.^ = ("' ^ - «'!/ + « 2/ - 2/ (X. (2/> "" yy ^h ^)y 
 the other two conditions are 
 
 0,y =^ <p^ay, and 0, 1/ = cj), a y 
 
(32) 
 
 putting ay for 2/ in the second of these It becomes (p^ay =: 
 <p^ a} y and this substituted in the first gives, 
 
 _ 1 L 
 
 whose solution is <p^y ^ ip-'-y — y) xAVy «2/> «''2/» "^^) 
 
 hence 
 
 and the general solution of the equation is 
 
 V^ (X, y) =e^x (y> «i/» "'.y> "'^) + 
 
 + s-'(a^i/-a^y + cc^—^)x,{yy ccy^ a^y, a^^,) + 
 
 + ("'i' — .V) K i^i «2/> «* 2/j «'2/) sin ^ + 
 I _i 
 
 + (a3 1/ - a y) X, (« 1/, «' ?/, a33', 7/) COS X, 
 
 (60), Given the equation xj. {x, y) = ^l^t^l^lll^ 
 
 where a is such a function that a" j: = x. 
 
 This equation may be reduced to the solution of the partial 
 differential equation 
 
 and the arbitrary functions of y which occur In Its solution, 
 must be determined by the conditions of the equation. 
 
 (6l). Given the equation 
 
 d \lr{a — X, y) _ d xjy (.r, b — y) 
 dy dx ^ 
 
 put a ^ X for Xy also h ^ y for ^, then we have the two 
 equations 
 
(33) 
 
 dyj^jx, y) _ _ (j y^ja •- Xf h - y) 
 dy dx 
 
 _ d\l^(a — x, b—y) _ dy\^{x, y) 
 dy dx 
 
 If the first of these be difFerentiated relative to y, and the 
 second relative to x ; then the right side of the first resulting 
 equation vi^Ill be identical with the left side of the second, 
 and we shall have 
 
 d" x^ (^r, y) ^ d'xl. (x, y) , 
 dy^ dx" 
 
 the solution of this partial differential equation is 
 
 ^ (J-, y)=(p{x + y) ^<p, {X - y) ; 
 
 the two arbitrary functions must be determined so as to 
 satisfy the equation ; we have 
 
 i±k^:lljl = ^' in- X + y)- ^'M-x - y\ 
 dy 
 
 ^^^'''^l'"^^ = ^'(6 +^-y) + f\(-i + ^ + y), 
 
 <p' and (p\ being the differential coefficients of (p and 0, these 
 two expressions must be identical, hence 
 
 <p' {a - X - y) = (p'(b + X - y), 
 and 
 
 - (p\ (a - ^' + y) = <p\ (- b + X + y), 
 the solutions of which equations are 
 
 and 
 
 <p\(^-^y) = {a-b-o,x-^y)xr { "^ -^ y, a -^ b -x-yl 
 and substituting these values, we have 
 
 tE 
 
(34) 
 
 (63). Given the equation 
 
 dx dy 
 
 Put -. for y and differentiate relative to .r, then 
 2/ 
 
 dx^ dx dy 
 
 Again, put- for r, and differentiate relative to i/, then 
 
 \x yy ^,^ d'x!^(x, y) 
 dxdy dy^ 
 
 hence 
 
 d' v^ {X, 1/) ^ X' d' yj, (X, y) 
 dy"" y"" dx'' ' 
 
 the solution of this equation of partial differentials is 
 x}^{x, 1/) = x(l> (^^J + (p, {xy) : 
 
 to determine the form of (/> and ^j, we have 
 d^(x') 
 
 = i^^^) + xy <p' {X y) + - 0\ (- ) , 
 dx y ^2/^ 
 
 ay x^y ^ocy/ x \x^ 
 
(35) 
 
 In order that these two expressions may coincide, we must 
 have 
 
 ^ (X J/) + xy <l>'(xy)=--l-^ <!>' ( J-) 
 
 The first of these multiplied by d{xy) may be put under the 
 form 
 
 whose integral is 
 
 xy<p{xy) = </>(—) 5 
 
 \xy/ 
 
 the solution of which functional equation is 
 
 ^/ xy V -^ ?/ ^ 
 the solution of the second equation is 
 
 ^y^ V X \y x^ 
 employing these values of <p and <p^, we have 
 
 v> 
 
 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<f)~'f<pX=:<p^'f''(l)X. 
 
 and yfy^'x zz(p-'f^(p(p-'f<P^ = 0~'/^0'^> 
 
 and the equation becomes 
 
 <p-'f<p{Q.a - 2) -(p-^p(px. 
 
 This equation may be satisfied in the following manner : by 
 making / a periodic function of the second order, we have 
 y« V = v, and the equation becomes 
 
 — 'y <^ (2 « — J) = </) — '/<^ T, 
 
 or 
 
 <p{^a - x) = <t>x. 
 
(37) 
 
 This is satisfied by making any symmetrical function of x 
 and 2 a — x\ As an example take fv = — v, also 
 
 <^ X = X . 2 a — a: —• 2 a X ■— ct", 
 then 
 
 and 
 
 y^x =zd>-' f<px = a ± \/ a^ , 
 
 ^ J ^ V x'' - ciax 
 
 i^m). Given x/^ (^i 1\ ^ ^i/ x 
 
 x/zT = (p — ^f(px where and f are determined by the 
 equations ^ 
 
 
 
 x = ;;^ < , X V and f^ x = x. 
 
 (67). Given \lr a x = \l^^ x, where a''v = v 
 putting y\r X = (p~~ ^ f<p x, we have 
 
 (p — 'f (pa X = (p^^f^ (p X 
 
 determine (p from the condition (p x = (p ax \ hence, 
 
 X = '^\ Xf a X, a" ~ ' 0? 
 
 and let /be such a function that f^x= x, then the equation 
 is satisfied. 
 
 (68). Given ^^ ax = xjy^ x where q >p and a'' x = x, 
 the substitution (p ~ ^f<p x instead of \/^ will give, 
 <p-'fp<pax = cp-^f^<p.x, 
 
 and this is satisfied if <p x — xi^y ^ ^y a'^ — 'x) 
 and also f'^—^v = v, for it then becomes 
 
 (p — ' fp (pa x=z(p~^fp(p .iy where (p x =z (pax. 
 
(38) 
 
 If in a function of two variables, as xjx {x, i/), we sub- 
 stitute the function itself insteaxi of one of those quantities, 
 the result is denoted thus, 
 
 ^ { V-(^, y\ y\ = V''{x, y\ 
 
 if the function itself is substituted simultaneously for x and 
 ^, it is denoted thus 
 
 (69). Given x/.^"^(x, ij) =^ a. 
 
 By means of the substitution <^ "" 'y x^ for y\r x, we are 
 enabled to reduce functional equations of any order to those 
 of the first, a substitution nearly resembling it, will be of 
 equal value for those which contain two or more variables, 
 by assuming 
 
 yl^ix, y) = <p-\f{(px, <py), 
 
 we have 
 
 x|,^U-, 3^) =</)-•/ i (/)-'/((/) ^', (/>!/), <p<p~\f{<px, <py)\ 
 
 = 0-y^(</,,,, 2/), 
 
 and substituting this value in the equation 
 
 cp-'/'^^cpX, <P7/) =a. 
 
 Put (p~\v for X, and ~ ' for y, also taking the function </> 
 on both sides 
 
 If therefore we are acquainted with a particular solution, 
 we find the general one ; let the function ^ 1- be tried, then 
 
 y 
 
139) 
 
 y 
 
 hence A = (pa, and the solution is 
 
 V^Cr, y) = ^-»(ti0«), 
 
 a variety of solutions may be found of difFerent forms, such 
 as 
 
 where a and /5 are any two homogeneous functions of the 
 same degree. 
 
 (70). If \/^ (j;, 1/) = « a? + byy 
 
 then y^^(^x^ y) =•• {a + bf-'{ax + 63^), 
 
 (71). If \}^ {x, y) is any homogeneous function of x, 
 and y of the degree «, 
 
 then 
 
 » y) = ! ^ ('*^? 
 
 Given v/.^a: 
 x/.(^, 1/) 
 
 Given 
 
 y^^ix, y) = 
 
 *~1 
 
 7/)}^ X {x/.(l, 
 
 , or-" • 
 
 (72). 
 
 , V) = \/v^ (^, 2/), 
 
 
 (73). 
 
 = ,/^^^ + ^^ 0(1). 
 
(40) 
 
 ^^'^^ (x + y),p(\) 20(f) 
 
 (74). Given V^'-'(:r, j/) = ]~^f'^l ' 
 
 ^0(1)-X»0(?) 
 1/0(1) +*^0(-) 
 
 (75). xj>~'(i-, 5,) = P^^(a^, ^), 
 
 provided a and /3 are homogeneous with respect to ^ and ^ ; 
 the first of the « + 1 degree, the second of the n^^, and also 
 at the same time a(l, 1) = /3(1, 1). 
 
 (76). Given x/.-"^ (^, t^) =^ F y^ {x, ^). 
 Another solution of the same equation is 
 
 ^^^'y^-'CM'^' 
 
 where a and /3 are two such functions, that when ^ = ?/, we 
 have also 
 
 a {X, y) = ^ (x, y), 
 {11), Given ^/.'~M^, ^) = ^/-(^,^), 
 
(41) 
 (78). Given ^'•'■{x, y) = \ V'(J-, ,V) 
 
 ^( 
 
 
 (79.) Given ^k'''(.i, y) = ! ^'■' (^, y) I ' 
 
 ^(..)=^<^±|^| 
 
 
 v/2 
 
 (80). Given x/x^'=» (.r, y) = - 
 
 ^{^^y^) = <p-' 
 
 (81). Given a x^''^(a-, y) = yV'' (^S ?y)» 
 put <p-'f((p oc, (py) for v// jc, then it becomes 
 
 X(p-'f'''{(px, (p7j)=y(p-'f'''(4>x, 4>y)i 
 putting </) - ' A' for .r, and (/) - ' 1/ instead of ?/, we have 
 
 if /-''(a:, ^) =1/, and /^'> (x, 3/) = :r, this equation, becomes 
 identical; but making /(r, 3/) = a - ^ -j/, these two equations 
 are verified ; consequently the general solution is 
 r/.(a:, 3^) =(p-'(a - (px - (py). 
 
 (82). Given v/^^'^ (r, 3/) . v^^'* (i", ?/) = ^'V* 
 (83). Given t vi^^'Cr, »/) = tf v/.''' (>r, ,v), 
 
Various methods for the solution of Functional Equations 
 may be found in the following writings : 
 
 Speculationes Analytico Geometricae, N. Fuss, Mem. de 
 rAcad. Imp. de St. Petersburg, Vol. IV. p. 225. 1811- 
 
 Memoirs of the Analytical Society, p. 96. 1813. 
 
 Observations on various points of Analysis, Phil. Trans. 
 J. F. W. Herschel, 
 
 Essay towards the Calculus of Functions, C. Babbage, 1815. 
 
 Ditto, Part II. p. 179. 1816. 
 
 Observations on the analogy which subsists between the 
 Calculus of Functions, and other branches of Analysis, 
 Phil. Trans. 1817. p. 197- C. Babbage. 
 
 Spence's Essays, 1819. Note by /. F. W. Herschel^ p. 151. 
 
 Annals of Philosophy, Nov. 1817. Mr. Horner, 
 
 Journal of the Royal Institution. C. Babbage, 
 
■ 
 
(42) 
 
 Various methods for the solution of Functional Equations 
 may be found in the following writings : 
 
 Speculationes Analytico Geometricae, N. Fuss. Mem. de 
 
 I 
 
 "■^'^>tr-^vvS:^-^ 
 
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