ODE/PDE- eighenvalues+ eigenfunctions

[Roni1985]

Homework Statement

it's already separable, so it's an ODE function.
X''+\lambda*X=0 0<x<1
X(0)=-2X(1)+X'(1)=0

Homework Equations

The Attempt at a Solution

this is a Sturm-Liouville eigenvalue problem.
Now, I know how to solve it and everything, but I'm not sure with one thing.

when I check the case where \lambda=0,
I get C2(-2x+1)=0
so C2 can be anything, correct ?
now what's my eigenfunction ?
X(x)=x is the eigenfunction ?

do I use the x=1/2 somewhere ?

Thanks.

 

[vela]

Homework Statement

it's already separable, so it's an ODE function.
X''+\lambda*X=0 0<x<1
X(0)=-2X(1)+X'(1)=0

Homework Equations

The Attempt at a Solution

this is a Sturm-Liouville eigenvalue problem.
Now, I know how to solve it and everything, but I'm not sure with one thing.

when I check the case where \lambda=0,
I get C2(-2x+1)=0

How did you get that?

so C2 can be anything, correct ?
now what's my eigenfunction ?
X(x)=x is the eigenfunction ?

do I use the x=1/2 somewhere ?

Thanks.

 

[Roni1985]

How did you get that?

well, when lamda is zero X''=0
so, X(x)=C1+C2*x
X(0)=C1=0
and by using the second BC, -2*C2*x+C2=0
so to get a nontrivial solution x=1/2, meaning C2 can be any number.

did I do something wrong ?

 

[vela]

The second BC is at x=1, so you get C2=0. So there's no non-trivial solution that satisfies the boundary conditions when λ=0.

 

[Roni1985]

The second BC is at x=1, so you get C2=0. So there's no non-trivial solution that satisfies the boundary conditions when λ=0.

shoot, you are right :\

forgot to plug in the 1.

Thanks for your help.