I have 2 solutions off my teacher which is right??

$\displaystyle 0<= x <= 1$ and $\displaystyle t > 0$

$\displaystyle
u_x (0,t) = u_x (1,t) = 0
$

for
$\displaystyle 0<= x <= 1$ and $\displaystyle t > 0$

My one general solution has At + b before the series and the other doesn't which is right.

Im guessing its due to lambda = 0 but do we need it or not