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) –...

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Extracted 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.