# Solution of the following PDE using European put option

I'm reading some articles about PDE and I found the following PDE, with $$q_1,A >0$$:
$$g_t(t,y)+ \beta^2yg_y(t,y)+\frac{1}{2}\beta^2y^2g_{yy}(t,y)-q_1 g(t,y)=0 \quad (t,y) \in [0,T), \times (0,+\infty)$$.
with boundary conditions $$\begin{cases} g(t,0)=F, t \in [0,T] \\ g(T,y)= \max\{F-Ay,S\} \end{cases}$$
The solution proposed is the following:
$$g(t,y)=Ae^{(\beta^2-q_1)(T-t)}p_{put}(t,y)+Se^{-q_1(T-t)}$$
where $$p_{put}$$ is the price of a European put option with strike price $$\frac{1}{A} (F-S)$$ in a BS market with volatility of risky asset $$\beta$$ and rate of riskless asset $$\beta^2$$.
My question is the following. I cannot understand actually the solution becasue if I substitute in the PDE it solves but it doesn't satisfy the condition $$g(t,0)=F$$ becasue I obtain $$g(t,0)=Fe^{-q_1(T-t)}$$. Can anyone help me?