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Auxiliary time dependent Schrodinger equation

Recall the relationship:
$$\phi(t) = C_+ exp[-i(\frac{E}{\hbar})t] + C_- exp[i(\frac{E}{\hbar})t]$$ .............(1)
Recall, from the present special case:
$$C_+ = 0$$ ........(2)
Substitute equ (2) into equ (1), we have that:
$$\phi(t) = C_- exp[i(\frac{E}{\hbar})t]$$ .......(3)
Also,  recall the relationship:
$$\Psi(r,t) = \psi(r) \phi(t)$$ ..........(4)
Substitute equ (3) into equ (4):
$$\Psi(r,t) = C_- exp \psi (r)  [i(\frac{E}{\hbar})t]$$ ..............(5)
Equ (5) can be differentiated with respect to t, and by rearranging we have:
$$\Psi(r,t) = -i(\frac{\hbar}{E}) \frac{d \Psi (r, t)}{dt}$$ ..............(6)
Recall the relationship:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = E \Psi(r,t)$$ ...........(7)
Substituting equ (6) into the right hand side of equ (7) we have:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = i\hbar \frac{d \Psi(r,t)}{dt}$$ .........(8)
Equ (8) is the auxiliary time dependent Schrodinger equation
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