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Im a bit stuck at part (e) and (f)

The 4 basic solutions are:

sinkx*coskct

sinkx*sinkct

coskx*coskct

coskx*sinkct

I got the 4 boundary conditions as:

y(0,t)=0

y(L,t)=0

y(x,0)=sin (2pi/l)x

dy/dt at t=0, =0

For part (e), I could eliminate coskx*coskct and coskx*sinkct from y(0,t)=0.

How do I choose between the other 2 solutions?

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for part (f)

How do I do it? Thanks.