So I have to find a transformation of $u$ that is of the form $\tilde u=f(t)u$ to reduce my problem
$u_t -ku_{xx} +(vu)_x +cu=0$
to
$\tilde u_t -k\tilde u_{xx} +a\tilde u_x=0$
I'm not really sure what to do, or what kind of problem it is to look it up either. I have a feeling I might have to use $u=e^{t\lambda}$ but that would leave me with $u_x$ and $u_{xx} =0$.
Do I have to substitute my value for $u$ and its derivatives into my original equation to find $\lambda$ and the do something else, I really don't know.
Thanks in advance
Just insert $u=\tilde u/f(t)$ in your equation $$u_t -ku_{xx} +(vu)_x +cu=0$$ $$\frac{f \tilde u_t - \tilde u f_t}{f(t)^2} - k \tilde u_{xx}/f(t) +(v \tilde u)_x/f(t) +c \tilde u/f(t)=0$$ $$\frac{f \tilde u_t - \tilde u f_t}{f} - k \tilde u_{xx} +(v \tilde u)_x +c \tilde u=0$$ $$\tilde u_t - \tilde u f_t/f - k \tilde u_{xx} +(v \tilde u)_x +c \tilde u=0$$ If $c=f_t/f$ ($c$ is a constant, right?), then $$\tilde u_t - k \tilde u_{xx} +(v \tilde u)_x=0$$ which has the desired form.
So you need $$\frac{df}{dt}=c f$$ $$f=\exp(c t)$$