= Solution
For the interior stationary candidate, let $\gamma=1/(\alpha+\beta-1)$. Its formal relation $\ell=1-\beta c/\alpha$ gives
$$
\dot x-rx=1-\frac{\alpha+\beta}{\alpha}Ce^{-\gamma rt}.
$$
Integration therefore gives $x=Ae^{rt}+Be^{-\gamma rt}-1/r$. Imposing $x(0)=x_0$ and $x(T)=0$ yields
$$
A=\frac{1-e^{-\gamma rT}(1+rx_0)}{r(e^{rT}-e^{-\gamma rT})},\qquad
B=\frac{(1+rx_0)e^{rT}-1}{r(e^{rT}-e^{-\gamma rT})}.
$$
This recovers the printed expression algebraically, but \b[does not turn its stationary control into an optimum]; it may also violate $\ell\ge0$.
The actual globally optimal wealth from part (ii) is
$$
\boxed{x(t)=
\begin{cases}
(x_0+r^{-1})e^{rt}-r^{-1},&0\le t\le t_*,\\[2mm]
Ce^{rt}\dfrac{e^{aT}-e^{at}}a,&t_*\le t\le T,
\end{cases}\qquad a=\frac{r\alpha}{1-\alpha}.}
$$
The budget equation makes the two branches agree at the switch. Direct differentiation gives $\dot x=rx+\ell-c$ on both branches, and $x(T)=0$. This solution remains nonnegative throughout, even though only the terminal nonnegativity constraint was imposed. It provides the complete corrected solution for $\alpha+\beta>1$.
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