Question

Find the eigenvalues and associated eigenfunctions of the
Strum-Liouville problem: xy''+y'+(λ/x)y=0
y'(1)=0,y'(e^(2π))=0.

Answer 1

$case \_1: \quad \lambda=-p^2<0\\ x^2y''+xy'-p^2y=0\Rightarrow y_1=x^p,y_2=x^{-p}\Rightarrow \\ y=c_1\cdot x^p+c_2\cdot x^{-p} \quad and \quad y'(1)=y'(e^{2\pi})=0\Rightarrow c_1=c_2=0.\\ case \_2: \quad \lambda=0\\ x^2y''+xy'=0 \Rightarrow y=c_1\cdot ln(x)+c_2 \quad and \quad y'(1)=y'(e^{2\pi})=0\Rightarrow c_1=0.\\ case \_3: \quad \lambda=p^2>0\\ x^2y''+xy'+p^2y=0\Rightarrow y=c_1sin(pln(x))+c_2cos(pln(x))\quad and \quad \\ y'(1)=y'(e^{2\pi})=0\Rightarrow\\ y'=\frac{p}{x}(c_1cos(pln(x))-c_2sin(pln(x)))\Rightarrow c_1=0,\quad p=\frac{k}{2},\quad k\in Z.\\ \lambda = (\frac{k}{2})^2,\quad y=c_2\cdot cos(\frac{k}{2}\cdot ln(x));$