Questions from chapter 2 of Zwiebach's A First Course In String Theory

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Hi,

I have a couple questions from chapter 2 of Zwiebach's A First Course In String Theory, second ed.

1) Zwiebach says [in the context of compact extra dimensions] "it seems very difficult, if not altogether impossible, to construct a consistent theory with more than one time dimension". Why is that? The best explanation my simple brain on Kruskal coordinates came up with is going faster than the speed of light is analogous to utilizing an extra time dimension to some extent, and (in the unlikely scenario I'm correct) if the converse were also true, then an extra time dimension would equate surpassing the speed of light. That being said, I can imagine constructing perfectly obedient metrics with extra negative signatures as easily as I can do so with positive signatures (is my terminology there correct?).

2) Zwiebach writes Schrodinger's equation as

E = -(\hbar2/2m)(1/ψ(x))(d2ψ(x)/dx2) -(\hbar2/2m)(1/ø(y))(d2ø(y)/dy2)

and derives

ψk(x) = cksin(k*π/a)

øl(y) = alsin(l*y/R) + blcos(l*y/R)

as solutions, and

Ek,l = (\hbar2/2m)[(k*π/a)2 + (l/R)2]

as the energy eigenvalues.

How did he derive those? The trig functions kinda came out of nowhere for me, and I'm unfamiliar with the concept of taking eigenvalues from (what appears to my untrained eye to be) a regular equation (i.e. not a matrix). Sorry for my stupidity here, I gather I probably don't have the appropriate prerequisites for the book, so if anyone has any resource suggestions to help me gain the appropriate background I would be grateful for the advice. Any hints on how I might resolve my questions are appreciated. Also, if anyone knows where I can access the solutions to the problems for the text that would be just awesome. I imagine it will be very difficult trying to glean adequate intuition from this book if all my answers are wrong but in my ignorance I assume they are correct.

Thanks to anyone who read all that, like I said hints appreciated.
 
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