A Riccati solution for Burgers’ equation

In the last few years a one-dimensional, time-dependent and nonlinear approximation to the Navier-Stokes equations, (1) below, has found applications in fields as diverse as number theory, gas dynamics, heat conduction, elasticity, etc. Probably the most important reason for this is that the complete and explicit solution of this equation became known in 1950. That solution, however, applies only to the homogeneous part of Eq. (1). In an attempt to tackle the nonhomogeneous case, we relate Eq. (1) to a Riccati equation, through a similarity transformation. Via this route, it is shown that solutions to the nonhomogeneous equation can be obtained.