The viscous Burgers' equation can be converted to a linear equation by the Cole–Hopf transformation,
which turns it into the equation
which can be integrated with respect to  to obtain
where  is an arbitrary function of time. Introducing the transformation  (which does not affect the function ), the required equation reduces to that of the heat equation
The diffusion equation can be solved. That is, if , then 
The initial function  is related to the initial function  by
where the lower limit is chosen arbitrarily. Inverting the Cole–Hopf transformation, we have
which simplifies, by getting rid of the time-dependent prefactor in the argument of the logarthim, to