Question

Find the characteristic values and characteristic functions of the storm lionville problem

d²y/dx²+λy=0, y(0)=0,y(π)=0,

where λ is negative

Answer 1

if $\lambda<0$ then $\lambda=-\mu^2$, where µ is real and non-zero. The general solution:

$y(x)=Ae^{\mu x}+Be^{-\mu x}$

for boundary conditions:

$y(0)=A+B=0\implies A=-B$

$y(\pi)=Ae^{\mu \pi}+Be^{-\mu \pi}=0\implies \mu=0$

So, there is only trivial solution with A=0,B=0:

$y(x)=0$