= Solution
Choose a reference epoch with $H_0=\dot a(t_0)/a_0\ne0$ and let $x=a/a_0$. Define the <critical density> and each <cosmological density parameter> by
$$
\rho_{\mathrm{crit},0}=\frac{3H_0^2}{8\pi G},\qquad
\boxed{\Omega_{i,0}=\frac{\rho_{i,0}}{\rho_{\mathrm{crit},0}},\qquad
\Omega_{K,0}=-\frac{K}{a_0^2H_0^2}.}
$$
For a closed universe $\Omega_{K,0}<0$. Take the dark-energy component to be a <cosmological constant>, as indicated by its $\Lambda$ label. The <constant-equation-of-state density scaling> gives
$$
\rho_r=\rho_{r,0}x^{-4},\quad\rho_m=\rho_{m,0}x^{-3},\quad
\rho_s=\rho_{s,0}x^{-2},\quad\rho_w=\rho_{w,0}x^{-1},\quad\rho_\Lambda=\rho_{\Lambda,0}.
$$
Substitution into the <Friedmann equation> gives
$$
\boxed{\frac{H^2}{H_0^2}=\Omega_{r,0}x^{-4}+\Omega_{m,0}x^{-3}+(\Omega_{s,0}+\Omega_{K,0})x^{-2}+\Omega_{w,0}x^{-1}+\Omega_{\Lambda,0}.}
$$
At the reference epoch, $\Omega_{r,0}+\Omega_{m,0}+\Omega_{s,0}+\Omega_{w,0}+\Omega_{\Lambda,0}+\Omega_{K,0}=1$. The <Friedmann acceleration equation> is
$$
\boxed{\frac{\ddot a}{aH_0^2}=-\Omega_{r,0}x^{-4}-\frac12\Omega_{m,0}x^{-3}+\frac12\Omega_{w,0}x^{-1}+\Omega_{\Lambda,0}.}
$$
Strings have $\rho_s+3P_s=0$ and make no direct acceleration contribution, although they affect the Friedmann constraint. The normalized density parameters are undefined at a static epoch, where $H_0=0$; the static solution must therefore be described directly with dimensional densities.
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