Uniqueness of sinusoidal functions for first order differential equations with constant shift I am wondering about solutions to the following differential equation:
$f(x)=C_1 \cdot f'(x+C_2) \; \forall x \in \mathbb{R} \; \exists \; C_1, C_2 \in \mathbb{R}$. With $C_1, C_2$ being constant. Are the solutions uniquely in the family of sin/cos functions? It bugs me that I was not able to come up with a counterexample except for the trivial solution $f(x)=0$. 
 A: for c2=0 you have the solution $$f(x)=A*e^{1/c_1}$$  with  c2!=0 you have $$f(x)=A*e^{1/c_1} -c2/c1$$ no sin or cos
you get sin or cos for second order equations
trula
A: With the help of the Laplace transform and assuming some simplifications as for instance
$$
f(x) = c_1\phi(x+c_2)f'(x+c_2)
$$
with $\phi(x)$ the Heaviside step function, an approximation can be worked out. After Laplace transforming we have
$$
F(s) = c_1\left(s e^{c_2 s}F(s)-f(c_2)\right)
$$
or
$$
F(s) = -\frac{c_1f(c_2)}{1-c_1 s e^{c_2 s}}
$$
now introducing the Padé approximation of first order for $|c_2| < 1$
$$
e^{c_2 s} \approx \frac{1+\frac{c_2 s}{2}}{1-\frac{c_2 s}{2}}
$$
we have for small $c_2$
$$
f(t)=\frac{f\left(c_2\right) e^{-\left(\frac{1}{2 c_1}+\frac{1}{c_2}\right)t} \left(\left(6 c_1+c_2\right)
   \sinh \left(\frac{\sqrt{4 c_1^2+12 c_2 c_1+c_2^2} t}{2 c_1 c_2}\right)-\sqrt{4 c_1^2+12
   c_2 c_1+c_2^2} \cosh \left(\frac{\sqrt{4 c_1^2+12 c_2 c_1+c_2^2} t}{2 c_1
   c_2}\right)\right)}{\sqrt{4 c_1^2+12 c_2 c_1+c_2^2}}
$$
