Stefan's Law of Radiation states that the radiation energy of abody is proportional to the fourth power of the absolute temperature T of a body. The rate of change of this energy ina surrounding medium of absolute temperature M is thus (dT)/(dt)=k(M^4 - T^4) where k > 0 is a constant. Show that the general solution of Stefan's equation is ln|(M+T)/(M-T)|+2tan^(-1)(T/M)=4M^3 kt+c where c is an arbitrary constant.

Stefan's Law of Radiation states that the radiation energy of abody is proportional to the fourth power of the absolute temperature T of a body. The rate of change of this energy ina surrounding medium of absolute temperature M is thus
$\frac{dT}{dt}=k\left({M}^{4}-{T}^{4}\right)$
where k > 0 is a constant. Show that the general solution of Stefan's equation is
$\mathrm{ln}|\frac{M+T}{M-T}|+2{\mathrm{tan}}^{-1}\left(\frac{T}{M}\right)=4{M}^{3}kt+c$
where c is an arbitrary constant.
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Lillianna Mendoza
$\frac{dT}{dt}=k\left({M}^{4}-{T}^{4}\right)$
$⇒\frac{dT}{{M}^{4}-{T}^{4}}kdt$
$⇒\int \frac{1}{\left({M}^{2}+{T}^{2}\right)\left({M}^{2}-{T}^{2}\right)}dT=\int kdt$
$⇒\frac{1}{{M}^{4}}\int \frac{1}{\left(1+\frac{{T}^{2}}{{M}^{2}}\right)\left(1-\frac{{T}^{2}}{{M}^{2}}\right)}dT=\int kdt$
We have,

Our equation becomes
$\frac{1}{2{M}^{4}}\int \left[\frac{1}{1-\frac{{T}^{2}}{{M}^{2}}}+\frac{1}{1+\frac{{T}^{2}}{{M}^{2}}}\right]dT=\int kdt$
$⇒\frac{1}{2}\int \left(\frac{1}{{M}^{2}-{T}^{2}}+\frac{1}{{M}^{2}+{T}^{2}}\right)dT=\int kdt$
Integrating both sides, we shall get