(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Compute the eigenvalues/functions of the given regular S-L problem

f''(x)+λf(x)=0

0<x<π

f(0)=0

f'(π)=0

2. The attempt at a solution

First off, why is π not included in the given boundary if it tells you f'(x) at π?

Now for my attempt:

assuming λ=0

f''(x)+0*f(x)=0

f''(x)=0

f'(x)=a, and f'(π)=0 so a=0

f(x)=ax+b, but a=0 so f(x)=b, and f(0)=0 so b also =0

therefore

f(x)=ax+b=0+0=0

so this doesn't really help

assuming λ>0

f''(x)+λf(x)=0

char eqn: r^2+λ=0

r=±√(λ)i

f(x)=c1*cos(√(λ)x)+c2*sin(√(λ)x)

f(0)=0 so plug in 0 ->f(0)=c1*cos(0)+c2*sin(0)c1=c1=0, therefore c1=0

now we are left with

f(x)=c2*sin(√(λ)x)

f'(x)=c2*√(λ)*sin(√(λ)x)

f'(π)=c2*√(λ)*sin(√(λ)π)=0

we can conclude than c2=0 or λ=0 but that wouldn't get us anywhere

instead, we assume

sin(√(λ)π)=0

√(λ)π=n*π (where n=1,2,...)

√(λ)=n

λ=n^2

so for λ>0 we have an eigenvalue of λ=n^2 where n=1,2,... and an eigenfunction of f(x)=c2*sin(nx) (plug n^2 into λ)

the last case is λ<0

f''(x)+λf(x)=0

char eqn: r^2+λ=0

r=±√(λ)

f(x)=c1*e^(√(λ)x)+c2*x*e^(√(λ)x)

f(0)=c1*e^0+c2*0*e^0=c1=0, therefore c1=0

we are left with

f(x)=c2*x*e^(√(λ)x)

f'(x)=c2*x*√(λ)*e^(√(λ)x)+c2*e^(√(λ)x)

f'(π)=c2*π*√(λ)*e^(√(λ)π)+c2*e^(√(λ)π)=0

we can factor it to make it a little more manageable

f'(π)=c2*e^(√(λ)π)*(√(λ)π+1)

once again, we can see that c2 and λ=0, but are trivial

assume that the last term =0

√(λ)π+1=0

√(λ)π=-1

√(λ)=-1/π

λ=1/π^2

(here, can I specify that only the negative root is used and not the positive one?)

so this gives us an eigenvalue of λ=1/π^2 and an eigenfunction of f(x)=c2*x*e^(±x/π)

The answer in the back of the book is

λ=(2n-1)^2 /4

f(x)=sin((2n-1)x/2) n=1,2.....

My answer looks nothing like that so what am I misunderstanding?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Basic sturm liouville boundary problems

**Physics Forums | Science Articles, Homework Help, Discussion**