 Solution

Put $q=1-2r/{\sigma }^2$. The Black-Scholes reflection symmetry maps $F$ to $(S/B{)}^{q}F(B^2/S,t)$, which solves the same PDE. Their difference vanishes at $S=B$. For $S>B$ at maturity the reflected argument is below $B$, where the terminal function in part ii is zero. Thus the boundary and terminal data of the alive down-and-out put are satisfied:The killed-domain Feynman-Kac formula identifies this solution with the discounted payoff conditional on survival. For $S\le B$, or after any prior barrier hit, the actual price is zero. The reflected difference is a formula on the alive domain $S>B$; extending that algebraic expression to $S<B$ generally gives a nonzero, possibly negative function and is not a valid barrier-option price.