= Solution
Write $L=\log\psi$, with $\psi>0$. Under the <Cole-Hopf transformation> $q=2\epsilon L_\theta$, direct differentiation gives
$$
q_Z-qq_\theta-\epsilon q_{\theta\theta}=2\epsilon\partial_\theta\{L_Z-\epsilon(L_{\theta\theta}+L_\theta^2)\}.
$$
Since $L_{\theta\theta}+L_\theta^2=\psi_{\theta\theta}/\psi$, <Burgers' equation> is equivalent to
$$
\partial_\theta\left(\frac{\psi_Z-\epsilon\psi_{\theta\theta}}\psi\right)=0.
$$
The ratio is therefore a function $a(Z)$ alone. Multiplying $\psi$ by $\exp[-\int a(Z)\,dZ]$ leaves $q$ unchanged and removes this scalar freedom. Thus one may choose the normalization for which
$$
\boxed{\psi_Z=\epsilon\psi_{\theta\theta}.}
$$
This is the <heat equation>. The positive sign in $q=2\epsilon\partial_\theta\log\psi$ corresponds to the negative conservative flux $-q^2/2$ used here; the more usual positive-flux convention has the opposite sign.
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