In Steven Shreve's excellent book, page 436, it says the forward rate $f(t,T)$ of the Heath-Jarrow-Morton (HJM) model explodes as $t\to T-$. The attached screenshot shows the calculation where the HJM dynamics is simplified to the deterministic case, that is supposed to be the dominant contribution in this limit, but the calculation is missing a factor of $T-t$. This factor is still there in the text right after Eq.(10.4.1), but is missing from the ODE in (10.4.2).
Question 1: I don't know if this was deliberately dropped or not. If yes, why?
One can actually solve the ODE even with that factor included $$ \frac{\partial f}{\partial t}(t,T) = \sigma^2(T-t)f(t,T)^2,\quad t\to T-. $$ Let $s=T-t$ and $F(s)=f(T-s,T)$ then $$ -\frac{F'(s)}{F(s)^2}=\sigma^2 s,\quad s\to0+. $$ The LHS is a total derivative and we get $$ F(s)=\left(\frac12\sigma^2 s^2+C\right)^{-1}, $$ where $C$ is an integration constant. This does explode for $s\to0$, as anticipated, but only if $C=0$.
Question 2: But I don't know how to argue about the choice $C=0$. Any ideas?
Regarding question 1, I am also not sure what happened to the $T-t$ term. Maybe it was just forgotten.
But with respect to question 2, I would suggest to use a concrete starting value $F(T)$ for time $s=T$. Then $C=1/F(T)-\frac{1}{2} \sigma^2 T^2$ and $$ F(s) = \frac{F(T)}{1+\frac{1}{2} \sigma^2 F(T) (s^2-T^2)} $$ (similar to Shreve‘s solution of the ODE without the $T-t$ term). This solution $F(s)$ explodes for time $s^2 = T^2 - \frac{2}{\sigma^2 F(T)}$.