I need to find the eigenvalues and eigenfunctions for the basic strum-liouville problem below with boundary conditions given:
$$y''+\lambda y=0$$ $$y(0)=0, y(2)=0 $$ on[0,2]. I know this is a basic SL form and problem but I am still quite confuse with the idea. This is what I did so far: $$y''+\lambda y=0 $$ $$m^2+ \lambda=0$$ $$y(x)=acos(\sqrt{\lambda} x)+bsin(\sqrt{\lambda} x)$$ Now using our boundary conditions: $y(0)=0=> acos(0)+bsin(0)=> a*1+0$ so a=0 and $y(2)=0=> acos(\sqrt{\lambda} 2)+bsin(\sqrt{\lambda} 2)=>bsin(\sqrt{\lambda} 2)$ because a=0 so $b \neq 0$ thus $$sin(\sqrt{\lambda} 2)=0$$
Don't know what to do from here to find the eigenvalues and eigenfunctions.
$\begin{array}{c} \sin \left( {\sqrt \lambda 2} \right) = 0 = \sin \left( {n\pi } \right)\\ \sqrt \lambda 2 = n\pi \\ \lambda = \frac{{{n^2}{\pi ^2}}}{4} \end{array}$
then, the eigenvalues are
${\lambda _n} = \frac{{{n^2}{\pi ^2}}}{4}$
now, the eigen functions are
${y_n} = \sin \left( {\frac{{n\pi }}{2}} \right)x$
I think this will be helpful for you.