Black-Scholes Formula Satisfies Black-Scholes Equation

My professor told me that each term of the Black-Scholes formula satisfy the Black-Scholes PDE. I have been trying to work this out for the last week and have been unsuccessful. I correctly showed the first term satisfied it, but I have had difficulty with the second term. My work is below:

We need to show that:

(where $N(w)$ is a standard normal distribution evaluated at $w$) satisfies the following:

Where $Q_s,\;Q_{ss}\;and\,Q_t$ are the derivative of Q w.r.t S, the second derivative of Q w.r.t S, and the derivative of Q w.r.t t respectively.

Let $d_2=\frac{log\big(\frac{S}{k}\big)+(r-\delta+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}-\sigma\sqrt{T-t}$


When I plug these derivatives into the Black-Scholes PDE, I cannot get the equality to hold. Can anyone please help?


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