$\displaystyle u = X(x) T(t) $

using seperation of variables

$\displaystyle \frac{X''}{x} = \frac{T'}{KT} = -\lambda $

$\displaystyle X'' + \lambdax = 0 $

$\displaystyle T' + \lambdaKT = 0 $

when $\displaystyle \lambda = 0 $

$\displaystyle X'' = 0 $

Am not sure how to proceed from here. if anyone can show or help. thank you.