Solve the functional equation $f(x+1)-f(x)=x*\sin(x) $

Question

Solve $f(x+1)-f(x)=x*\sin(x) $

What have you tried so far? – ArsenBerk Jun 09 '19 at 19:46 — ArsenBerk, Jun 09 '19 at 19:46
@ArsenBerk calculus of finite differences – MorrisonFJ Jun 09 '19 at 19:48 — MorrisonFJ, Jun 09 '19 at 19:48

score 4 · Answer 1 · answered Jun 09 '19 at 19:59

4

Take any function $f(x)$ defined on $[0,1)$ and then continue to all $x$ using the equation.

answered Jun 09 '19 at 19:59

colt_browning

1

Bonus: If $f$ is continuous on $[0,1)$ and $\lim_{x\to1^-}f(x)=f(0)$, then the continuoation will be continuous on all of $\Bbb R$. – Hagen von Eitzen Jun 09 '19 at 20:14
@HagenvonEitzen but $ f(x)=1/4 csc^2(1/2) (sin(x)-x (sin(1-x)+sin(x)))$ . I don't know how to come to this decision – MorrisonFJ Jun 09 '19 at 20:17
@colt_browning but f(x)=1/4csc2(1/2)(sin(x)−x(sin(1−x)+sin(x))) . I don't know how to come to this decision – MorrisonFJ Jun 09 '19 at 20:30
@MorrisonFJ Why do you think it's true? Because it's the answer given in the textbook? Well, if this is known to be the correct answer, then the question is apparently missing something. Perhaps the condition that the function must be analytical? This would make the problem a lot harder. – colt_browning Jun 09 '19 at 20:35

1 Answers1