merton model alternative derivation

[david paich]

Suppose

(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

and follow
processes

,

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

,

satisfy the Black-Scholes equation