PDE separation of variables Hi could someone guide me this problem
It says ,
$ u_t - u_{xx}-2 u_x=0  $
Use the method of separation of variables to find all possible solutions. 
Could someone help me out for this problem. I'm beginner at PDE. I would be much appreciated if you able to show some partial work so that I can understand.
Thanks in advance for taking my consideration

If $u(x,t) = X(x)T(t)$, then 
$$\frac{T'}{T} = \frac{X''}{X} + 2 \frac{X'}{X}$$
LHS depends on $t$ only, RHS depends on $x$ only, so both sides are equal to a constant, say $\lambda$.  You can take it from here...
