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 $\displaystyle f'(x)=f(x+1)$ have any nontrivial solutions?

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

3. i think that
$\displaystyle k=ln \; k$
there is no solution.

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

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

Solving

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

gives if I am not mistaken

$\displaystyle w=\pm 1$

$\displaystyle tg^2 \; a =-1$
there is no solution.

4. Originally Posted by Danny
I believe it does. If you seek solutions of the form $\displaystyle f(x) = c e^{kx},$ then we obtain that $\displaystyle k$ satisfies $\displaystyle k = e^k$.
Yeah but $\displaystyle x<e^x$ 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 $\displaystyle f'(x)=f(x+1)$ have any nontrivial solutions?
If $\displaystyle \mathcal {L} \{f(t)}\}= \varphi(s)$, for the basic properties of the Laplace Tranform is...

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. If it is not requested that $\displaystyle f^{'}(t) = f(1+t)$ 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...

$\displaystyle f(t)= e^{t}$ , $\displaystyle 0<t<1$

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

$\displaystyle f(t)=\left\{\begin{array}{ll}\sin 2 \pi t ,\,\,0 < t <1\\{}\\ \cos 2 \pi t ,\,\, 1 < t <2\\{}\\ -\sin 2 \pi t,\,\, 2<t<3 \\{}\\ -\cos 2 \pi t,\,\, 3<t<4\end{array}\right.$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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