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Wave equations separation of variables

Would appreciate some help with the above question.

For part a, i believe i correctly determined the separable solutions to be

X=cos(αx) or sin(αx)
T=cos(αct) or sin(αct)
Where alpha is just a constant

So a linear combination of these would be

y(x,t)=(Acos(αx) + Bsin(αx))(Ccos(αct) + Dsin(αct))

However when I apply the conditions given I can't seem to get any of the constants out, Im not sure if the solution I've found is wrong or if im not applying the conditons correctly, I would appreciate any help.

Thanks.
Original post by Eremor
Would appreciate some help with the above question.

For part a, i believe i correctly determined the separable solutions to be

X=cos(αx) or sin(αx)


Doesn't the second boundary condition restrict this to X=sin⁡xX=\sin x, so that you have:


So a linear combination of these would be

y(x,t)=(Acos(αx) + Bsin(αx))(Ccos(αct) + Dsin(αct))

y(x,t)=sin⁡x(Acos⁡αct+Bsin⁡αct)y(x,t)=\sin x(A \cos \alpha ct + B \sin \alpha ct) ?

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