= Solution
Promote the canonical field and momentum $\pi=f'$ to operators satisfying the <canonical commutation relation>, $[\hat f(\tau,\mathbf x),\hat\pi(\tau,\mathbf y)]=i\delta^{(3)}(\mathbf x-\mathbf y)$. A real field has Fourier reality condition $\hat f_{\mathbf k}^\dagger=\hat f_{-\mathbf k}$. Its oscillator expansion is
$$
\boxed{\hat f_{\mathbf k}=u_k\hat a_{\mathbf k}+u_k^*\hat a_{-\mathbf k}^\dagger}.
$$
The <creation and annihilation operators> add or remove excitations in the selected mode basis, with
$$
[\hat a_{\mathbf k},\hat a_{\mathbf k'}^\dagger]=\delta^{(3)}(\mathbf k-\mathbf k'),\qquad
[\hat a_{\mathbf k},\hat a_{\mathbf k'}]=[\hat a_{\mathbf k}^\dagger,\hat a_{\mathbf k'}^\dagger]=0,\qquad
\hat a_{\mathbf k}|0\rangle=0.
$$
Canonical normalization requires the <Wronskian> $u_ku_k^{*\prime}-u_k^*u_k'=i$.
For the <Bunch-Davies vacuum>, choose the positive-frequency short-wavelength behavior $u_k\sim e^{-ik\tau}/\sqrt{2k}$ as $k|\tau|\to\infty$. For the proposed solution, direct differentiation gives
$$
u_k'=\frac{e^{-ik\tau}}{\sqrt{2k}}\left[-ik-\frac1\tau+\frac{i}{k\tau^2}\right],\qquad
u_k''=\frac{e^{-ik\tau}}{\sqrt{2k}}\left[-k^2+\frac{ik}\tau+\frac2{\tau^2}-\frac{2i}{k\tau^3}\right].
$$
Substitution cancels every term in $u_k''+(k^2-2/\tau^2)u_k$. Its <Wronskian> is $i$, and its large-$|k\tau|$ limit is the required flat-spacetime positive-frequency mode. Thus
$$
\boxed{u_k=\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)}.
$$
A term with the conjugate frequency is another allowed classical solution, but would describe a different quantum state, through a <Bogoliubov transformation>. Its coefficient is set to zero by the early-time vacuum condition, not by absence of a second solution to the differential equation.
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