In this article, first, a method of successive approximations for the treatment of the unsteady thermal boundary layer in laminar flow is composed. Then, the analytical solutions of these differential equations for two laws of the motion of a body: $u_e=At^\alpha U_e(s,z)$, $\alpha\geq0$, $u_e=Ae^{ct}U_e(s,z)$, $c>0$, are found, Finally, the possiblity to facilitate the calculation of the thermal boundary layer in the case $u_e=At^\alpha U_e(s,z)$, using the simpler solutions for the case $u_e=Ae^{ct}U_e(s,z)$, is elaborated.