Cole-Hopf transformation for $u_{t}+c_{1}c_{2} (u_{x})^{2}=c_{1} u_{xx}$ I came across ,in some paper, that the
 $$u_{t}+c_{1}c_{2} (u_{x})^{2}=c_{1} u_{xx}$$

can be solved via the Cole-Hopf transformation. Is there a generalized Cole-Hopf transformation? I tried plugging in 
$$u=2c_{1}\frac{f_{x}}{f}$$

and got 

$$f^{3}(f_{t}-c_{1}f_{xx})_{x}+f^{2}f_{x}(c_{1}f_{xx}-f_{t})+2c_{1}^{2}c_{2}(f_{xx}f-(f_{x})^{2})^{2}=2c_{1}f_{x}f(f_{x}^{2}-ff_{xx})$$

Thanks

Also, what if $c_{1}(t),c_{2}(t)$ are functions of $t$ and not constants?
Hint:

Let $\begin{cases}x_1=x\\t_1=\int_k^tc_1(\tau)c_2(\tau)~d\tau\end{cases}$ ,

Then $u_x=u_{x_1}(x_1)_x+u_{t_1}(t_1)_x=u_{x_1}$

$u_{xx}=(u_{x_1})_{x_1}(x_1)_x+(u_{x_1})_{t_1}(t_1)_x=u_{x_1x_1}$

$u_t=u_{x_1}(x_1)_t+u_{t_1}(t_1)_t=c_1(t)c_2(t)u_{t_1}$

$\therefore c_1(t)c_2(t)u_{t_1}+c_1(t)c_2(t)(u_{x_1})^2=c_1(t)u_{x_1x_1}$

$u_{t_1}+(u_{x_1})^2=\dfrac{u_{x_1x_1}}{c_2(t)}$
The Cole-Hopf transfomation for the PDE
$$
u_t - a \Delta u + b |D u|^2 = 0
$$
is $w = e^{-b u/a}$, which, in your case, is $ w = e^{-c_2 u}$. Performing this transformation on your equation, leads to
$$
w_t - c_1 w_{xx} = 0,
$$
a standard heat equation. You'll have to be careful with the boundary conditions, though.