Periodic functions (or similar)

[zetafunction]

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

[tex] y(x)=y(kx) [/tex] they are 'periodic' but under DILATIONS

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

[tex] axy'(x)+bx^{2}y''(x)=e_{n}y(x) [/tex]

if the Lie Group is of translations [tex] y(x+1)=y(x) [/tex] 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:
[tex]
bx^{2}y''+axy'-e_{n}y=0
[/tex]
Look for solutions of the for, [itex]y(x)=x^{n}[/itex], then:
[tex]
bn(n-1)x^{n}+anx^{n}-e_{n}x^{n}=0\Rightarrow (bn(n-1)+an-e_{n})x^{n}
[/tex]
So to obtain solutions we look for solutions of the quadratic:
[tex]
bn^{2}+(a-b)n-e_{n}=0
[/tex]
Which gives:
[tex]
n=\frac{b-a\pm\sqrt{(b-a)^{2}+4be_{n}}}{2b}
[/tex]