Time evolution of the radius of a hot gas bubble inflated by a wind from an early SMBH accreting at its Eddington limit (energy-driven case). The following parameters are assumed: f_w = 0.05, λ = 1, M₀ = 10⁴M_⊙ (hence L_w,0 = 6.3 × 10⁴⁰ erg s^-1), ϵ = 0.1, ρ_gas = 10^-28 g cm^-3. The black solid curve shows the numerical solution obtained for the equation of motion (Eq. (13)) assuming an exponentially growing wind with L_w(t) = L_w,0e^t/t_Sal and assuming R,v → 0,0 for t → 0. The analytic approximation to the numerical solution (Eq. (16)), is shown by the solid green line. The purely exponential (but unphysical) solution given by Eq. (14) is shown by the black dotted curve. The black dotted line shows the solution obtained for a constant energy source L_w(t) = L_w,0 (Eq. (17)). The vertical dotted lines mark the Salpeter time t_Sal and the time t₉ needed to grow the SMBH to 10⁹M_⊙ as labelled. The bubble radius scales as t^{3 / 5} for t ≪ t_Sal and e^{t/ (5t_Sal)} for t ≫ t_Sal. The red points show the results of a low-resolution simulation run with the hydrodynamical code RAMSES assuming the same input parameters. The results of the simulation are in excellent agreement with the numerical solution to Eq. (13).