  The first star-forming objects which formed at high redshifts during the
cosmic dawn (CD) also emitted photons between Lyman-$\alpha$ and Lyman-limit
frequencies. These photons are instrumental in coupling the spin temperature of
the neutral hydrogen (HI) atoms with the kinetic temperature of the
intergalactic medium (IGM). Along with this coupling effect, these photons also
impact the kinetic temperature by exchanging energy with the HI atoms. The
injected Lyman-$\alpha$ photons in general cool the medium, while the continuum
photons heat the medium. While studying this effect in the literature,
quasi-static profile around the Lyman-$\alpha$ frequency is assumed. In this
paper, we solve the time-dependent coupled dynamics of the photon intensity
profile along with the evolution of the thermal state of the IGM and HI spin
temperature. It is expected that, during the CD era, the IGM has a mix of
continuum photons with 10-20% of injected photons. For this case, we show that
the system reaches thermal equilibrium in around 1 Myr, with final temperature
in the range 50-100 K. This time scale is comparable to the source lifetime of
PopIII stars at high redshifts. One impact of switching off short-lived sources
is that it can keep the system heated above the temperature of the quasi-static
state. We also show that the quasi-static equilibrium for the continuum photons
is only achieved on time scales of 100 Myr at $z\simeq 20$, comparable to the
age of the Universe. We also briefly discuss how the Lyman-$\alpha$ induced
heating can impact the 21 cm signal from CD.
