So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation.
Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve.
I can do this successfully to obtain :
$Ae^{+i\sqrt{{2mE}/{\hbar^{2}}}x}+Be^{-i\sqrt{{2mE}/{\hbar^{2}}}x}$
My lecturer has a small section titled :
Solving for the Free Schrodinger Equation
$$V=0$$
$$\frac{\hbar^{2}}{2m}\frac{\partial^2\psi}{\partial x^2}+E\psi=0$$
$$E=\frac{p^2}{2m}$$
$$\psi=Ce^{-{iEt}/{\hbar}+{ipr}/{\hbar}}$$
This is the solution to the free TISE and TDSE.
He seems to be doing the same thing as me initially but he's obtained a different result ?
Also, the first section of his answer :$e^{-{iEt}/{\hbar}}$ is the solution to the time part of the equation (described above).