Change of variables in Black-Scholes equation

[Traveler0714]

The general form of BS equation looks like:

(dV/dt) + 0.5 * (sigma)^2 * S^2 * (d2V/dS2) + r * S * (dV/dS) - r * V =
0

Then change the variables: F = e^( r * (T-t)) * S and V(S,t) = U(F,t)

The book gives me the answer as in the following:

(dU/dt) + 0.5 * (sigma)^2 * F^2 * (d2U/dF2) - r * U = 0

I have tried several times, but can't reproduce the solution. Can
anyone help me on the derivation of the solution?

Thanks in advance.

[actuaryalfred]

How did you justify the change of variables?

A standard derivation could be found in

http://www.rotman.utoronto.ca/%7Ehull/Technical%20Notes/TechnicalNote7.pdf