black scholes - Merton model riskless self-financing derivation

Suppose $dA_t = A_t[\mu dt+\sigma dW_t]$ (assets' value) under the physical measure, plus the other assumptions of the Merton model.

Suppose further that debt and equity are tradeable assets that satisfy $A_t = D_t+E_t$ and follow processes $D_t = D(t,A_t)$, $E_t = E(t,A_t)$ for differentiable functions.

By considering a locally risk-free self-financing portfolio of bonds and equity(which by necessity will earn the risk-free rate of return), prove directly that both $D$, $E$ satisfy the Black-Scholes equation:

$$\partial_t f+\frac{1}{2}\sigma^2 A^2 \partial_A^2 f+r A \partial_A f-r f = 0$$