Question

(PDE)(Sturm-Liouville) I'm solving an SLDE, and I'm not sure if I'm doing part of this right; I think I'm getting a trivial solution where I shouldn't be. Posted below in a second.

Answer 1

\[y''-\lambda y=0, \ \ \ y(0)=y(1)=0\]

Answer 2

What I've done: \[\text{Let $\lambda=-\alpha^2$, where $\alpha$ is a real constant. }\]\[y_h=c_1e^x+c_2e^-x, \ \ \ y'_h=c_1e^x-c_2e^{-x}\]\[y(0)=c_1+c_2=0.\]\[y(1)=c_1e+\frac{c^2}{e}=0.\]

Answer 3

\[(1) \ \ \ c_1+c_2=0\]\[(2) \ \ \ c_1-\frac{c^2}{e}=0\]
(Note, the y(1) in the second post should be the same as the one in this one)

Answer 4

Substituting (1) into (2) in the form of c_1=-c_2 or vice versa, I just end up with \[c_1\bigg(e+\frac{1}{e}\bigg)=0\]

Answer 5

Is this wrong, or do I need to look for other values of lambda? (e.g. zero, +alpha^2) @Kainui

Answer 6

Are you sure you've written the problem correctly? I thought for a Sturm Liouville equation you have this form \[\Large \frac{d}{dx} \left( p(x) \frac{dy}{dx} \right) + q(x)y+ \lambda r(x) y =0\] with p(x), q(x), and r(x) all greater than 0. It appears as if your q(x)=0 and r(x)<0 but p(x)=1 is completely fine. Another thing, you have not really used the first derivative in your boundary conditions, which is why I thought it was odd you calculated it right before using your boundary conditions, but didn't plug it into anything, so I'm not really sure what's going on there.

Answer 7

Maybe somehow that negative sign will get absorbed into the eigenvalue based on how you've done it, but still something seems odd since having your coefficients both come to 0 seems to be either wrong or very simple solution. =P

Answer 8

That does have the general form of an SLDE if q(x)=0, p(x)=1, and r(x)=1. It's silly, but yeah, it does it the general form. Hmm.

Answer 9

Sure, but I'm saying q(x) needs to be larger than 0, I guess I need to review this method.

Answer 10

Sure thing, I'm reading it directly from the book and what you've said makes sense to me, too, but when I've brought up this exact same SLDE in a different context on OS and asked why this fit the general form, someone explained it to me as fitting the form as such.

But yeah, I don't know what I'm doing, lol.

Answer 11

Brb, riding bike to different area, ~15 mins.

Answer 12

ya i do remember having to set some part of it to 0 so that it looks nice

Answer 13

i dontt thnk its q(x) though

Answer 14

in your case it is since that part doesnt exist