For different n, sin(λ_n x) are linear independent to each other, and if they add up to zero, we must have every coefficient to be zero. This is another fundamental result in ODE.

a (0, 1) + b (1, 0) = (0, 0)

Then a = b = 0 must hold. That\’s the same reasoning.

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Use the initial condition u_t(x, 0) = 0, what do you get?

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It won’t take me much time. I just think it would be better if you can work it out yourself. That’s something very fundamental in ODE theory, and worth practicing.

Below is the proof that you can refer to:

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There is no difference whether it is X or X’

The general solution is X(x) = a_n sin(λ_n x) + b_n cos(λ_n x)

You plug in X(0) to see that b_n must be all zero,

then you plug in X\'(L) = 0 to get λ_n

Is that clear to you? Or you still want me to write down specific process?

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Do you know how to solve ODE with boundary condition?

Namely, if y” = λy?  and y(0) = y(L) = 0?

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