# Thread: Fibonacci differential equation

1. ## Fibonacci differential equation

Hello!
This is just something I've asked myself, and not something I need help with for school.

Does the differential equation $f'(x)=f(x+1)$ have any nontrivial solutions?

2. I believe it does. If you seek solutions of the form $f(x) = c e^{kx},$ then we obtain that $k$ satisfies $k = e^k$.

3. i think that
$
k=ln \; k
$

there is no solution.

$
f(x)=sin(wx+a)
$

$
f'(x)=w \; cos(wx+a)
$

Solving

$
w \; cos(wx+a)=sin(w(x+1)+a)
$

gives if I am not mistaken

$
w=\pm 1
$

$
tg^2 \; a =-1
$

there is no solution.

4. Originally Posted by Danny
I believe it does. If you seek solutions of the form $f(x) = c e^{kx},$ then we obtain that $k$ satisfies $k = e^k$.
Yeah but $x always! So there's no solution of that form. What about other forms?

5. Originally Posted by Bruno J.
Hello!
This is just something I've asked myself, and not something I need help with for school.

Does the differential equation $f'(x)=f(x+1)$ have any nontrivial solutions?
If $\mathcal {L} \{f(t)}\}= \varphi(s)$, for the basic properties of the Laplace Tranform is...

$\displaystyle \mathcal {L} \{f^{'}(t)}\}= s\ \varphi(s) - f(0)$

$\displaystyle \mathcal {L} \{f(1+t)}\}= e^{s}\ \varphi(s)$ (1)

Now from the (1) it follows that a solution of the equation $f^{'} (t) = f(1+t)$ is [if any inverse LT exists]...

$\displaystyle f(t)= \mathcal{L}^{-1} \{\frac{c}{s-e^{s}}\}$ (2)

... being $c=f(0)$ a constant...

Kind regards

$\chi$ $\sigma$

6. If it is not requested that $f^{'}(t) = f(1+t)$ is valid for every t but for 'almost every t' we have several 'candidate functions'. One of the most simple is the periodic function with period T=1 defined as...

$f(t)= e^{t}$ , $0

Very suggestive is also the 'PSK modulated' periodic function with period T=4 defined as...

$f(t)=\left\{\begin{array}{ll}\sin 2 \pi t ,\,\,0 < t <1\\{}\\ \cos 2 \pi t ,\,\, 1 < t <2\\{}\\ -\sin 2 \pi t,\,\, 2

Kind regards

$\chi$ $\sigma$

7. Originally Posted by Bruno J.
Yeah but $x always! So there's no solution of that form. What about other forms?
What about complex $x$? Big Picard's gives us infinitely many solutions.