 Solution

At the first half-temperature time, the solution of Newton's law of cooling givesSince $\alpha >2$, this time is positive and finite. The instantaneous temperature increase supplies the new initial condition $T(t_1+)=\alpha T_0/2+\beta$. Restart the same first-order linear differential equation at $t_1$:This is Newton cooling with an instantaneous temperature jump. The time translation in the exponential decay is essential: its prefactor is the excess immediately after the jump.