 Solution

The Cole-Hopf transformation for the sign convention here is $Q=q_x/q$, with $q\ne 0$ and $q_t=q_{xx}$. Indeed $q_t/q=Q_x+Q^2$, soConversely a solution $Q$ determines a locally nonzero $q$ through $q_x=Qq$, $q_t=(Q_x+Q^2)q$; their compatibility is this Burgers equation. A spatially constant term can be removed by rescaling $q$ in time.

Set $\mu =q\psi$ in the heat-equation pair. Division by $q$ givesTheir compatibility can be checked without referring back to $q$: writing $A=ik-Q$, $B=-Q_x-Q^2-k^2$, we have ${\psi }_x=A\psi +1$, ${\psi }_t=B\psi +Q+ik$. The coefficient of $\psi$ in ${\psi }_{xt}-{\psi }_{tx}$ is $-Q_t+Q_{xx}+2Q{Q}_x$, and the constant term is $(ik-Q)(Q+ik)-Q_x-B=0$. Thus this linear auxiliary pair is compatible exactly for the required viscous Burgers equation. As in part (a), adjoining a constant component turns it into a homogeneous $2\times 2$ matrix Lax pair.