2.3.7.3 ✓ Mathematica. Time used: 1.657 (sec). Leaf size: 44

ode=(T[t]+1/Sqrt[t^2-T[t]^2])*D[T[t],t]== T[t]/(t*Sqrt[t^2-T[t]^2])-t; 
ic={}; 
DSolve[{ode,ic},T[t],t,IncludeSingularSolutions->True]
 

\begin{gather*} \text {Solve}\left [-\arctan \left (\frac {\sqrt {t^2-T(t)^2}}{T(t)}\right )+\frac {t^2}{2}+\frac {T(t)^2}{2}=c_1,T(t)\right ] \end{gather*}