Solutions to the heat equation, spatial or time decay? In reading standard texts there seems to be two standard solution to the heat equation:
$$\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$$
The first is obtained when assuming a real separation constant (if solving by separation of variables) this gives you a wave of the form: 
$$e^{-t}\sin(x)$$
The second is obtained when you allow for a complex separation constant and this gives you a solution of the form:
$$e^{-x}\sin(t-x)$$
I'm becoming very confused as to which solution to use in which scenario, I have answered two separate questions which seem to be the same but each one is looking for a different solution. 
Could anybody help me work out which one is appropriate and when?
 A: There are not two solutions, but two class of solutions. Which class to use depends on the boundary and initial conditions.
A: Given
$$u_t=a u_{xx}$$
coupled with initial condition and homogeneous (spatial) boundary conditions.
assume $u= X(x)T(t)$ then
$$XT'=a X''T$$
divide by $a u$ to get
$$\frac1a\frac{T'}T = \frac{X''}X$$
since a function of $t$ equals to a function of $x$ for any $x,t$ can only be when both are constant functions one write
$$\frac1a\frac{T'}T = \frac{X''}X=\lambda$$
next you solve $$X''-\lambda X=0$$ together with homogeneous boundary conditions to determine (countable set of) eigenvalues $\lambda_n$ and eigenfunctions $X_{\lambda_n}\equiv X_n$ . Then for each $\lambda_n$ one solves $$T'-a\lambda_n  T=0$$ to for a general solutions $T_n$.
Finally one writes $$u=\sum_n A_n X_n T_n$$
and use initial condition to determine $A_n$, some time the solution has closed form.
The function $e^{-x}\sin(t-x)$ doesn't solve any problem with homogeneous boundary condition. In order to solve such  problem one solve auxiliary inhomogeneous problem ($v_t=a v_{xx}+f(x,t)$) with homogeneous boundary conditions.
