Question

Find the characteristic values and the characteristic functions of the sturn_ liouville problem.

d/dx[x(dy/dx)]+(λ/x)y=0

y¹(1)=0,y¹(e^2x)=0 where λ is nonnegative

Answer 1

if $\lambda=0$ :

$xy'=c_1$

$y=c_1lnx+c_2$

$y'(1)=c_1=0$

$y=c_1$

if $\lambda>0$ :

$\lambda=k^2$ with $k>0$

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

an Euler equation with indicial equation:

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

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

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

$y'(1)=c_2=0$

$y'(e^{2x})=-c_1sin(2kx)/e^{2x}=0$

$2kx=\pi n$

$k=\pi n/(2x)$

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

$y_n=cos(\pi nlnx/2x)$