8.9.12 f(x) – f(y) = (x – y)f' () This functional equation can be rewritten to the form f(2) – {(1) = r (**"). x + y (8.451) %3D x - y 424 Difference Equations Note that if we set x = y + h, then ( flu + h) – f(w) – r (y+;). f' (y + 2 (8.452) 0, we have the identity f'(y) = f'(y). and upon taking Lim h Now, taking the derivatives of x + y ( (a) – f(1y) = (x – 9)s" ("). (8.453) respectively, with x and then y, gives (ro)-r(뿌) -» (3)r(부) x + y x + y '() = ' ()+ (* - u) () " (*) (8.454) %3D and (-)()-)-() • (9) --wr(*) x + y fl" x + y x + y (x – y) f" (*+y (8.455)
8.9.12 f(x) – f(y) = (x – y)f' () This functional equation can be rewritten to the form f(2) – {(1) = r (**"). x + y (8.451) %3D x - y 424 Difference Equations Note that if we set x = y + h, then ( flu + h) – f(w) – r (y+;). f' (y + 2 (8.452) 0, we have the identity f'(y) = f'(y). and upon taking Lim h Now, taking the derivatives of x + y ( (a) – f(1y) = (x – 9)s" ("). (8.453) respectively, with x and then y, gives (ro)-r(뿌) -» (3)r(부) x + y x + y '() = ' ()+ (* - u) () " (*) (8.454) %3D and (-)()-)-() • (9) --wr(*) x + y fl" x + y x + y (x – y) f" (*+y (8.455)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:8.9.12 f(x) – f(y) = (x – y)f' (
)
x+y
This functional equation can be rewritten to the form
f(x) – f(y)
x + Y
(8.451)
x – Y
424
Difference Equations
Note that if we set x =
y + h, then
( (u + h) – {(u) – g' (v+5).
f' (y +
(8.452)
and upon taking Lim h – 0, we have the identity f'(y) = f'(y).
Now, taking the derivatives of
x + y
( f(æ) – f(y) = (x – y)f" ( "5"),
(8.453)
2
respectively, with x and then y, gives
»(4) ~()
x + y
x + y
f'(x) = f'
2
+ (x – y)
(8.454)
2
and
/-(G)r(플)- (6) r() (3) -o/m (")
x + y
x + y
x + Y
f'
2
0 =
4
2
() «
x + y
(= – 9) f" (").
(8.455)
2
Therefore, f(x) is determined by the equation
C o"(2) = 0,
(8.456)
which has the solution
C(2) = Aa? + Bx +C,
f (x)
(8.457)
where (A, B, C) are arbitrary constants.
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