Question

Find the characteristics values and the characteristics functions of the sturn_liouville problem

d/dx[x(dy/dx]+(λ /x)y=0,

y¹(1)=0, y¹(e(2π))=0 where λ>0

Answer 1

if $\lambda>0$ :

$\lambda=k^2$ with $k>0$

$x^2y''+xy'+k^2y=0$

an Euler equation with indicial equation:

$r^2+k^2=(r-ik)(r+ik)=0$

$y=c_1cos(klnx)+c_2sin(klnx)$

$y'=-c_1sin(klnx)/x+c_2cos(klnx)/x$

$y'(1)=c_2=0$

$y'(e^{2\pi})=-c_1sin(2k\pi)/e^{2\pi}=0$

$2k\pi=\pi n$

$k= n/2$

$\lambda_n=(\ n/2)^2$

$y_n=cos( nlnx/2)$