= Solution
The <Cole-Hopf transformation> for the sign convention here is $Q=q_x/q$, with $q\ne0$ and $q_t=q_{xx}$. Indeed $q_t/q=Q_x+Q^2$, so
$$
Q_t=\partial_x(Q_x+Q^2)=Q_{xx}+2QQ_x.
$$
Conversely 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 $q$ gives
$$
\boxed{\psi_x+(Q-ik)\psi=1,\qquad\psi_t+(Q_x+Q^2+k^2)\psi=Q+ik.}
$$
Their 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 $-Q_t+Q_{xx}+2QQ_x$, 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 $2\times2$ <matrix> <Lax pair>.
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