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# Sturm-Liouville form question watch

1. Firstly the question asks me to solve the ODE:

y''+2gy'+(g^2 + n^2)y=0 subject to the boundary condition that the solution vanishes at x=o and x=pi. (where 'g' is a constant)

I get the solution to be yn=Asinnx

Next the questions asks to put the ODE into Sturm-Liouville form and hence find the weight function, w(x) such that:

INT(from 0 to pi) [ym(x)yn(x)w(x)dx]=0 for m 6=n

After putting the ODE into Sturm-Liouville form I get:

(exp(2gx)y')'+(n^2)exp(2gx)y+(g^ 2)exp(2gx)y=0

This suggests that w(x)=gexp(2gx) but I don't really see how to proceed from there, I can see that it might have something to do with the orthogonality of sine and cosine. Also, I don't not completely sure about what the question means by m 6=n, does it mean 6m=n?

Thanks.

P.s. I don't know how to use latex so sorry about the dodgy integral expression.
2. Since y_n is the solution to y''+n^2 y = 0, I don't really see how it can be a solution to the DE you posted. g is a constant, but you can't simply ignore it.
3. (Original post by DFranklin)
Since y_n is the solution to y''+n^2 y = 0, I don't really see how it can be a solution to the DE you posted. g is a constant, but you can't simply ignore it.
Hmm, when I solved the ODE there was a exp(-g) term which I just took into the constant A because g is a constant, is this incorrect?
4. I think you'll find it's an exp(-gx) term, which is not a constant.
5. (Original post by DFranklin)
I think you'll find it's an exp(-gx) term, which is not a constant.
/facepalm. Thanks.

Does the second part look correct however?

Ah, the exp term i missed out cancels out the w(x) in the integral and you're just left with an integral of sinnxsinmx.

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