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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.

â€¢ 'â€¢'â€¢â€¢

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 â€” x) = a â€” (aâ€”x)=za â€” a + x=:x.

If a(x) =â€” L_,

then a^X=:a{ax) = I == ^ ~ "^ = ^ " ^ ,

1 1 â€” A? â€” 1 X

I - X

and

- Â» ax â€” I 1

a' a: = a* a a: = = _

a. X

= 1 â€” I â€” 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 = \// â€” '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 â€” a =: 0, \l^ x s= a â€” 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=aâ€”x is a particular

solution therefore

\!y X = F{Xy xf/^ x) â€” X,

or

x â€¢{â– 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 =: v = i;>

hence the solution of the equation is

yjy X =z (p~^f(pX,

one solution is and hence ^x z=. (h â€” ^ ( J

more particular cases are

y^X = y^ X =.

a â€” X \ â€” 'd X

a" , \/a ar^ - a"

\I/^X== -â€” \jyX=:

a c â€” c^ X ' X

, ax â€” d^ . 1

\lrX=: -4^ X =:

1 - X

^"^^ (^zr^y >/^^=-iog(i-.aO

(a a?" â€” a^) ^

yf^x = V _>Â» 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 =i

1 +X/.X 1 + .^(-x)

put \/^i a^ == â€¢ â€” thus the equation becomes

^lr^x + X yp^i (^â€” x) = 1, and changing x into â€” a; we have

4^1 { â€” X) â€” ar\/^, (a) = 1, by which eliminating \l^{ â€” x) from

the former, we find

^' 1 + X^'

hence

yjyyX 1 â€” X

ylr X =

I â€” \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 ]â€”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:lÂ£) = 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 â€” 1

1 â€” a

if we take for fr its value â– , 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^ â€” \- h,

which becomes the given equation when a = l and ^ = 0.

By putting _. for x this is changed into

â– 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 then x^. ( 1 - :r = X-L-_-/ ;

and the equation becomes

^1^^' + ^.(1 â€”x) = 1,

whose general solution is

\/^i jc = ^- â€” ;

0-r+0(l-x)'

hence .

putting 1 â€” .r for x, and eliminating v/^ (1 â€” x), we find

[0(1-^) + Â«^^r

(24). Given (^|. xf + (^ -)' = ^LÂ±_f! v^ x . v/. f! â€¢,

^ X ^ X^ X

a"

divide by v/^ a . \^ â€” then

^x f_ _ X* + a*

~~aF ^l^ X '^

putting \'/i .r â€” ^ 3 this becomes

(20)

^l^ + ^^ â€” = â€” - â€” >

a particular solution of which is ^/^l ^ = x*; hence

= .1"^ or v/ X =L ^^ -v// â€”

, Â« X

X

and the general solution of this is

\^-x^\

(25). Given riLÂ±f + x ^ = 1 -f ^%

v/^ X -i- .2?

put \^, x = â€” , then the equation becomes

X

^, X + x'' x/., i = 1 + a:%

whose solution is >i/^x â€¢=. ^ â€” , ^-Iâ€” - ;

X

, a? + ^x ix'' + 1)0. r

hence ^ = ^^â€” ;

putting - for x and eliminating -v/^ - , we find

.^x= ; i â€” ; â€” 'â€”rr^^^ I +'"*")

X ^ 1/

(21)

(26). Given (v/. xf' . {^|r - xT - (^ ^T â€¢ (^ - ^)'" = ^ .r ;

putting >//, X = (^^ j:)'^ . (Vx â€” xf, it becomes

^/-.^ - ^^.(- ^)= '^^>

whose solution is ^t v = x + x (^> ~ '^)>

hence

and by the process for eliminating yjy {â€” x), we shall find

I \ x{^Â» '^T)-^r ] '^J

(27). Given y^rx â– {- fx ,^ o.x = /Â» x, where ax \^ such

a function of x that a'' a = cT ; putting Â« x for i', we have

â– y\rs/fx

^{fo.x) derived from the equation /a: ,fax-=:\y we have

\/{fa x) . Vx .r + x/(/^) . >/^ Â« X =/, (.r, a ,r),

which is a symmetrical equation, whose general solution is

,, r \ 1 fti'V, ax) .

^.^

1 ^^-l'

I â€” X X

hence

Y, +

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 "" yy ^h ^)y

the other two conditions are

0,y =^ "'^) +

+ s-'(a^i/-a^y + cc^â€”^)x,{yy ccy^ a^y, a^^,) +

+ ("'i' â€” .V) K i^i Â«2/> Â«* 2/j Â«'2/) sin ^ +

I _i

+ (a3 1/ - a y) X, (Â« 1/, Â«' ?/, 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 â€” X, y) _ d xjy (.r, b â€” y)

dy dx ^

put a ^ X for Xy also h ^ y for ^, then we have the two

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.

â€¢ 'â€¢'â€¢â€¢

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 â€” x) = a â€” (aâ€”x)=za â€” a + x=:x.

If a(x) =â€” L_,

then a^X=:a{ax) = I == ^ ~ "^ = ^ " ^ ,

1 1 â€” A? â€” 1 X

I - X

and

- Â» ax â€” I 1

a' a: = a* a a: = = _

a. X

= 1 â€” I â€” 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 = \// â€” '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 â€” a =: 0, \l^ x s= a â€” 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=aâ€”x is a particular

solution therefore

\!y X = F{Xy xf/^ x) â€” X,

or

x â€¢{â– 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 =: v = i;>

hence the solution of the equation is

yjy X =z (p~^f(pX,

one solution is and hence ^x z=. (h â€” ^ ( J

more particular cases are

y^X = y^ X =.

a â€” X \ â€” 'd X

a" , \/a ar^ - a"

\I/^X== -â€” \jyX=:

a c â€” c^ X ' X

, ax â€” d^ . 1

\lrX=: -4^ X =:

1 - X

^"^^ (^zr^y >/^^=-iog(i-.aO

(a a?" â€” a^) ^

yf^x = V _>Â» 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 =i

1 +X/.X 1 + .^(-x)

put \/^i a^ == â€¢ â€” thus the equation becomes

^lr^x + X yp^i (^â€” x) = 1, and changing x into â€” a; we have

4^1 { â€” X) â€” ar\/^, (a) = 1, by which eliminating \l^{ â€” x) from

the former, we find

^' 1 + X^'

hence

yjyyX 1 â€” X

ylr X =

I â€” \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 ]â€”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:lÂ£) = 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 â€” 1

1 â€” a

if we take for fr its value â– , 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^ â€” \- h,

which becomes the given equation when a = l and ^ = 0.

By putting _. for x this is changed into

â– 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 then x^. ( 1 - :r = X-L-_-/ ;

and the equation becomes

^1^^' + ^.(1 â€”x) = 1,

whose general solution is

\/^i jc = ^- â€” ;

0-r+0(l-x)'

hence .

putting 1 â€” .r for x, and eliminating v/^ (1 â€” x), we find

[0(1-^) + Â«^^r

(24). Given (^|. xf + (^ -)' = ^LÂ±_f! v^ x . v/. f! â€¢,

^ X ^ X^ X

a"

divide by v/^ a . \^ â€” then

^x f_ _ X* + a*

~~aF ^l^ X '^

putting \'/i .r â€” ^ 3 this becomes

(20)

^l^ + ^^ â€” = â€” - â€” >

a particular solution of which is ^/^l ^ = x*; hence

= .1"^ or v/ X =L ^^ -v// â€”

, Â« X

X

and the general solution of this is

\^-x^\

(25). Given riLÂ±f + x ^ = 1 -f ^%

v/^ X -i- .2?

put \^, x = â€” , then the equation becomes

X

^, X + x'' x/., i = 1 + a:%

whose solution is >i/^x â€¢=. ^ â€” , ^-Iâ€” - ;

X

, a? + ^x ix'' + 1)0. r

hence ^ = ^^â€” ;

putting - for x and eliminating -v/^ - , we find

.^x= ; i â€” ; â€” 'â€”rr^^^ I +'"*")

X ^ 1/

(21)

(26). Given (v/. xf' . {^|r - xT - (^ ^T â€¢ (^ - ^)'" = ^ .r ;

putting >//, X = (^^ j:)'^ . (Vx â€” xf, it becomes

^/-.^ - ^^.(- ^)= '^^>

whose solution is ^t v = x + x (^> ~ '^)>

hence

and by the process for eliminating yjy {â€” x), we shall find

I \ x{^Â» '^T)-^r ] '^J

(27). Given y^rx â– {- fx ,^ o.x = /Â» x, where ax \^ such

a function of x that a'' a = cT ; putting Â« x for i', we have

â– y\rs/fx

^{fo.x) derived from the equation /a: ,fax-=:\y we have

\/{fa x) . Vx .r + x/(/^) . >/^ Â« X =/, (.r, a ,r),

which is a symmetrical equation, whose general solution is

,, r \ 1 fti'V, ax) .

^.^

1 ^^-l'

I â€” X X

hence

Y, +

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 "" yy ^h ^)y

the other two conditions are

0,y =^ "'^) +

+ s-'(a^i/-a^y + cc^â€”^)x,{yy ccy^ a^y, a^^,) +

+ ("'i' â€” .V) K i^i Â«2/> Â«* 2/j Â«'2/) sin ^ +

I _i

+ (a3 1/ - a y) X, (Â« 1/, Â«' ?/, 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 â€” X, y) _ d xjy (.r, b â€” y)

dy dx ^

put a ^ X for Xy also h ^ y for ^, then we have the two

1 2

Online Library → Charles Babbage → Examples of the solutions of functional equations → online text (page 1 of 2)