Lost on eigenvalues

[Chris_K]

I'm having trouble getting started on this problem... I just really don't understand what to do.


Solve
X'+2X'+(\lambda-\alpha)X=0, 0<x<1
X(0)=0
X'(1)=0

a. Is \lambda=1+\alpha an eigenvalue? What is the corresponding eigenfunction?
b. Find the equation that the other eigenvalues satisfy.


I appreciate any help you can give me!

Thanks,
Chris

[ReyChiquito]

are u sure that the edo is correct?

you have there a linear second order d.o with constant coeficients and initial values, so your solution will be some linear comb. of exp[rt] (you have to calculate r of course).

i dont really understand why would you end with an eigenvalue problem in this way but maybe im not well informed.

[Chris_K]

Yeah, everything on there is correct. I'm not sure what you mean though...


nm... figured it out.

[ReyChiquito]

X''+2X'+(\lambda-\alpha)X=0

let X=e^{rt} implies

r^2+2r+(\lambda-\alpha)=0

so

r=-1\pm\sqrt{1-(\lambda-\alpha)}

X(t)=Ae^{(-1+\sqrt{1-(\lambda-\alpha)})t}+Be^{(-1-\sqrt{1-(\lambda-\alpha)})t}

X(0)=A+B

so A=-B[/tex]

X'(1)=A[r_{+}e^{r+}-r_{-}e^{r_{-}}]=0

[itex]A=0 would lead to the trivial solution, so

r_{+}e^{\sqrt{1-(\lambda-\alpha)}}=r_{-}e^{-\sqrt{1-(\lambda-\alpha)}}

\lambda=1+\alpha is clearly an eigenvalue

and \lambda_{m} satisfy the equation

\frac{-1+\sqrt{1-(\lambda_{m}-\alpha)}}{-1-\sqrt{1-(\lambda_{m}-\alpha)}}e^{2\sqrt{1-(\lambda_{m}-\alpha)}}=1