I'm trying to find a solution to: $v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$
$v(0, t) = 0$
$v_x(l, t) = 0$
$v(x, 0) = −U$
I have:
$v(x,t) = X(x)T(t)$
$X(x)T'(t) - kT(t)X''(x) = 0$
$\frac{T'(t)}{kT(t)} = \frac{X''(x)}{X(x)} = -\lambda$
$X''(x) = -\lambda X(x)$
$\delta^2 = \lambda$
$X(x) = C\cos(\delta x) + D\sin(\delta x)$
$X'(x) = -C\delta\sin(\delta x) + D\delta\cos(\delta x)$
$X(0) = C+ D = 0 => C=-D$
$X'(l) = -D\delta\sin(\delta l) + D\delta\cos(\delta l) = D\delta[\sin(\delta l)+\cos(\delta l)] = 0$
This is wear I keep getting suck
The correct value of $\delta$ is suppose to be $\frac{n + (1/2)}{l}\pi$ but I don't see how that is possible