Need help-eigenvalue problem

Hey!!!

Knowing that:
"The eigenvalue problem

Ly=(py')'+qy, a \leq x \leq b

is a Sturm-Liouville problem when it satisfies the boundary conditions:

p(a)W(u(a),v^*(a))=p(b)W(u(b),v^*(b))

,where

W

is the wronskian."

I have to show that the eigenvalue problem

y''+λy=0

, with boundary conditions

y(0)=0, y'(0)=y'(1)

is not a Sturm -Liouville problem.

This is what I've done so far:

Let

u, v^*

solutions of the eigenvalue problem

y''+λy=0

, then:

u(0)=0, u'(0)=u'(1)

and

v^*(0)=0, v^{*'}(0)=v^{*'}(1)

.

W(u(0),v^*(0))=u(0)v^{*'}(0)-u'(0)v^*(0)=0

W(u(1),v^*(1))=u(1)v^{*'}(1)-u'(1)v^*(1)=u(1) v^{*'}(0)-u'(0)v^*(1)

How can I continue? How can I show that this is not equal to 0?

Reply 1

Smaug123

15

Original post by mathmari

Hey!!!

W(u(1),v^*(1))=u(1)v^{*'}(1)-u'(1)v^*(1)=u(1) v^{*'}(0)-u'(0)v^*(1)

How can I continue? How can I show that this is not equal to 0?

Could you not find

u,v

and show it that way? Seems like cheating, but I don't see why you shouldn't do it.

Reply 2

mathmari

OP

1

Original post by Smaug123

Could you not find

u,v

and show it that way? Seems like cheating, but I don't see why you shouldn't do it.

I have found the eigenfunctions of the eigenvalue problem.They are

y_n(x)=\sin(2 n \pi x)

.If we suppose that

u

and

v^{*}

are solutions of the eigenvalue problem,do I have to take two different

n

at the eigenfunction to find

u

and

v

? I got stuck right now.. :confused:

Reply 3

Smaug123

15

Original post by mathmari

I have found the eigenfunctions of the eigenvalue problem.They are

y_n(x)=\sin(2 n \pi x)

.If we suppose that

u

and

v^{*}

are solutions of the eigenvalue problem,do I have to take two different

n

at the eigenfunction to find

u

and

v

? I got stuck right now.. :confused:

Hmm, now I'm confused. Because it's not a Sturm-Liouville problem, we aren't guaranteed to have a complete set of eigenfunctions, so we can't proceed by expressing u and v as a sum of eigenfunctions. I'm stumped, I'm afraid :frown:

Reply 4

mathmari

OP

1

Original post by Smaug123

Hmm, now I'm confused. Because it's not a Sturm-Liouville problem, we aren't guaranteed to have a complete set of eigenfunctions, so we can't proceed by expressing u and v as a sum of eigenfunctions. I'm stumped, I'm afraid :frown:

Oh ok... How did you mean it that I could find

u

and

v

?

Reply 5

Smaug123

15

Original post by mathmari

Oh ok... How did you mean it that I could find

u

and

v

?

In hindsight, I was assuming that any u and v could be expanded in terms of eigenfunctions. That's true for Sturm-Liouville problems, but not necessarily for non-SL ones.

Reply 6

mathmari

OP

1

Original post by Smaug123

In hindsight, I was assuming that any u and v could be expanded in terms of eigenfunctions. That's true for Sturm-Liouville problems, but not necessarily for non-SL ones.

Aha! Ok! Do you have any other idea what I could do?

Need help-eigenvalue problem

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