Hi everybody,
I am studying for an exam in financial mathematics, and have been going through past exam papers. Here is one of the questions I've come across, and I'm struggling with it:

The first part I have done, by direct susbtitution. After factoring out \frac{S^x}{S_d^x}, and messing around a little bit, I got something very similar to the Black-Scholes equation except the coefficient of the \frac{\partial V_1}{\partial S} term is slighly different. My PDE seemed reasonable.
So I proceed to part ii), where I struggle. I tried changing the variable of differentiation by using \frac{d \epsilon}{dS} = \frac{-S_d^2}{S^2} but my calculations don't work out, I think my differentiation has messed up somewhere along the way.
Could anyone please help me with this? I need to make sure I understand and can do this question, in case something similar comes up in the exam!
Many thanks,
lando45