 Solution

For the interior stationary candidate, let $\gamma =1/(\alpha +\beta -1)$. Its formal relation $\ell =1-\beta c/\alpha$ givesIntegration therefore gives $x=A{e}^{rt}+B{e}^{-\gamma rt}-1/r$. Imposing $x(0)=x_0$ and $x(T)=0$ yieldsThis recovers the printed expression algebraically, but does not turn its stationary control into an optimum; it may also violate $\ell \ge 0$.

The actual globally optimal wealth from part (ii) isThe 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 $\alpha +\beta >1$.