[imath]\displaystyle \frac{\partial Y}{\partial t} \neq Z'(t)[/imath]

Why do you want to do that?

If you want to use [imath]\displaystyle Z'(t)[/imath], the lemma should give us [imath]\displaystyle X'(t)[/imath]. Since the lemma didn't say anything about [imath]\displaystyle X'(t)[/imath], you cannot use [imath]\displaystyle Z'(t)[/imath].

The example should define [imath]\displaystyle Y = F(Z,t) = e^{-\mu t}Z[/imath].

Following the lemma will give us these results:

[imath]\displaystyle F_X = F_Z = \frac{\partial Y}{\partial Z} = e^{-\mu t}[/imath]

[imath]\displaystyle F_{XX} = F_{ZZ} = \frac{\partial^2 Y}{\partial Z^2} = 0[/imath]

[imath]\displaystyle F_t = \frac{\partial Y}{\partial t} = -\mu e^{-\mu t} Z = -\mu e^{-\mu t} Z(t)[/imath]

[imath]\displaystyle a(X,t) = a(Z,t) = \mu Z = \mu Z(t)[/imath]

[imath]\displaystyle b(X,t) = b(Z,t) = \sigma [/imath]