Periodic functions (or similar)

[zetafunction]

are there non-connstant function that satisfy the following asumptions ??

$$y(x)=y(kx)$$ they are 'periodic' but under DILATIONS

and also satisfy the differential equation of the form (eigenvalue problem)

$$axy'(x)+bx^{2}y''(x)=e_{n}y(x)$$

if the Lie Group is of translations $$y(x+1)=y(x)$$ we may have sine and cosine , however for the case of DILATIONS i do not know what functions can we take.

 

[hunt_mat]

Your differential equation is of Euler type:
$$bx^{2}y''+axy'-e_{n}y=0$$
Look for solutions of the for, $y(x)=x^{n}$, then:
$$bn(n-1)x^{n}+anx^{n}-e_{n}x^{n}=0\Rightarrow (bn(n-1)+an-e_{n})x^{n}$$
So to obtain solutions we look for solutions of the quadratic:
$$bn^{2}+(a-b)n-e_{n}=0$$
Which gives:
$$n=\frac{b-a\pm\sqrt{(b-a)^{2}+4be_{n}}}{2b}$$