(PDE)(Sturm-Liouville) I don't understand how in this document, in example 6.10, they got the general solution for lambda less than zero. Posted below shortly.

Question

mendicant_bias · Answer

(Page 3, Example 6.10) http://www.macs.hw.ac.uk/~robertw/F13YT2/chap6.pdf

mendicant_bias · Answer

How did they get that the general solution for $$y''=k^2y$$equals$$y(x)=A \cosh(kx)+B \sinh(kx)?$$

mendicant_bias · Answer

(When I say Page 3, I mean 3 in the PDF, not by the page numbers in the document)

perl · Answer

thats a fun exercise

perl · Answer

we want to solve  the equation
y ' ' - k^2 y = 0 
you can use power series method

mendicant_bias · Answer

Is this Froebenius(sp?) Method?

perl · Answer

i dont think we would use that here. 
we can use power series. i just did a problem like this yesterday

mendicant_bias · Answer

(Lol I don't remember Power Series method at all, so I might be a while on figuring this out)

perl · Answer

here http://openstudy.com/study#/updates/54f2e40fe4b0b9f2c71fef94

mendicant_bias · Answer

Thanks! Looking at some other problems for now (trying to focus on understanding Sturm-Liouville, it's highly unlikely that I'll have to solve a DE using a series method in this upcoming exam), but I will look at that later.

perl · Answer

sure

usukidoll · Answer

oh gawd I remember doing this.

usukidoll · Answer

let me guess.....going crazy on pdes?

usukidoll · Answer

what they did is that they  say suppose lamdba is less than 0 then we have a negative value such that if we have lambda = -k2 and plug it back into the equation, we will see that there are complex roots involved because the equation doesn't factor nicely...sorry for typos. I'm full and bed time soon.

usukidoll · Answer

in my honest opinion... pdes suck. So glad I passed it, but NEVER AGAIN! Wave equations are easy though.

mendicant_bias · Answer

I don't see how using the Power Series Method would result in me getting sinh or cosh, still.

I understand if exponential functions appeared somehow in the power series solution, that is possible, but that isn't the case, no?

usukidoll · Answer

hmm I'm trying to remember what I did... I did remember taking three cases when lambda is less than 0, 0, and greater than 0. But, I have never ever used Power Series in them either.