How to solve $(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$

Question

$$(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$$

I don't have any ideas about the solution of this problem. How can I solve this differential equation?

Answer 1

You told that there is symmetry with the minus sigh, which is correct.

Notice that $f'(x-1)+f'(x+1)=[f(x-1)+f(x+1)]'$, thus $\frac{f(x-1)+f(x+1)}{f(x)}=c$ yield $f(x)[f'(x-1)+f'(x+1)]-f'(x)[f(x-1)+f(x+1)]=0$.

Is it helpful?

Edit: the above is true if $f(x) \neq 0$. We need to check separately if $f(x)=0$ is a solution and it is.

Answer 2

Notice that $f(x)=0$ is a solution. Now assume $f(x)\neq0$ and we can divide both sides by $f^2(x)$ to get: $(\frac{f(x+1)}{f(x)})'+(\frac{f(x-1)}{f(x)})'=0$

$c f(x)=f(x+1)+f(x-1)$ And now we use characteristic polynomial to solve this recurrence:

$x^2-cx+1=0$ Hence if $c\neq2$

$f(x)=a (\frac{c+\sqrt{c^2-4}}{2})^x+b (\frac{c-\sqrt{c^2-4}}{2})^x$

And if $c=2$

$f(x)=ax+b$