 Solution

Let $D_K(S,t)=P_d(S,t)$ and $\gamma =1-2r/{\sigma }^2$. The Black-Scholes reflection symmetry gives another solution $(S/B{)}^{\gamma }{D}_K(B^2/S,t)$. Subtracting it enforces absorption:At $S=B$ the terms agree. At expiry the reflected argument satisfies $B^2/S>B>K$, so its digital-put payoff vanishes; the first term supplies the required terminal payoff. These facts and the bounded pricing solution's uniqueness prove the formula. The up-and-out digital put price is zero once knocked out.