Right heres a Q on the heat equation that should be simple but im making a mistake somewhere.....

Since I wasn't entirely sure what to do i figured if i sub in the values of a and b into the exponential then differentiate with respect to $\displaystyle \tau$ and again but twice with respect to x then both answers should be equal.

but for $\displaystyle U_{\tau}$ i got $\displaystyle -(\frac{1}{4} + \frac{r}{\sigma^2} + \frac{r^2}{\sigma^4})e^{ax + b\tau}$

and for $\displaystyle U_{xx}$ i got $\displaystyle \frac{1}{4} - \frac{r}{\sigma^2} + \frac{r^2}{\sigma^4} e^{ax + b\tau}$

have i made a mistake somewhere or should i have subbed $\displaystyle V(s,t) = e^{ax + b\tau} U(x, \tau)$ into the Black-Scholes equation (in which case i think ill leave this Q alone for now...)