= Solution
Write $\mathcal H=a'/a$ for the <conformal Hubble parameter> and $\theta_N=i\mathbf k\cdot\mathbf v_N$ for the <velocity divergence> of a <Fourier mode>. Work in the <synchronous gauge> comoving with the <cold dark matter>, so $\mathbf v_C=0$. This remains consistent with its pressureless momentum equation $\mathbf v_C'+\mathcal H\mathbf v_C=0$. Its density equation then gives $\delta_C'=-h'/2$.
For the string component, $1+3w_S=0$. Thus it makes no contribution to the trace gravitational source in the supplied perfect-fluid model. Substitute $h'=-2\delta_C'$ and $h''=-2\delta_C''$ in that trace equation to obtain
$$
\boxed{\delta_C''+\mathcal H\delta_C'-\frac32\mathcal H^2\Omega_C\delta_C=0.}
$$
The absence of a pressure-gradient term reflects the zero <cosmological sound speed> of the cold matter.
For the $w_S=-1/3$ <coasting fluid>, the continuity and momentum equations become
$$
\delta_S'=-\frac23\theta_S+\frac23\delta_C',\qquad
\theta_S'+2\mathcal H\theta_S+\frac12k^2\delta_S=0.
$$
The second equation follows by taking $i\mathbf k\cdot$ the vector equation; in particular, $i\mathbf k\cdot i\mathbf k=-k^2$, which fixes the pressure sign. Differentiate the first equation and use $\theta_S=\delta_C'-3\delta_S'/2$:
$$
\delta_S''=\frac43\mathcal H\delta_C'-2\mathcal H\delta_S'
+\frac13k^2\delta_S+\frac23\delta_C''.
$$
Eliminating $\delta_C''$ with the cold-matter equation gives
$$
\boxed{\delta_S''+2\mathcal H\left(\delta_S'-\frac13\delta_C'\right)
-\frac13k^2\delta_S-\mathcal H^2\Omega_C\delta_C=0.}
$$
These are equations for the specified idealized barotropic string fluid. A <cosmic string network> with additional <anisotropic stress> need not obey the same pressure response.
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