Solve the functional equation

Dec 2015
1,085
169
Earth
\(\displaystyle \prod_{i=1}^{n} \dfrac{d^{i-1} s(t/i)}{dt^{i-1} } =\dfrac{1}{n!}\).

\(\displaystyle s(t)\cdot s^{(1)}(\dfrac{t}{2})\cdot \dotsc \cdot s^{(n)}(\dfrac{t}{n})=\dfrac{1} {n!}\).
 
Dec 2015
1,085
169
Earth
\(\displaystyle

\dfrac{s(t)\cdot s^{(1)}(\dfrac{t}{2})\cdot \dotsc \cdot s^{(n+1)}(\dfrac{t}{n+1})}{
s(t)\cdot s^{(1)}(\dfrac{t}{2})\cdot \dotsc \cdot s^{(n)}(\dfrac{t}{n})}=s^{(n+1)}(\dfrac{t}{n+1})=1/n
\).

\(\displaystyle s(\dfrac{t}{1+n})=\dfrac{t^{n+1}}{n}\).

\(\displaystyle s(t)=\dfrac{(tn+n)^{n+1}}{n}\).