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:
<br /> bx^{2}y&#039;&#039;+axy&#039;-e_{n}y=0<br />
Look for solutions of the for, y(x)=x^{n}, then:
<br /> bn(n-1)x^{n}+anx^{n}-e_{n}x^{n}=0\Rightarrow (bn(n-1)+an-e_{n})x^{n}<br />
So to obtain solutions we look for solutions of the quadratic:
<br /> bn^{2}+(a-b)n-e_{n}=0<br />
Which gives:
<br /> n=\frac{b-a\pm\sqrt{(b-a)^{2}+4be_{n}}}{2b}<br />