= Solution
At the first half-temperature time, the solution of <Newton's law of cooling> gives
$$
1+(\alpha-1)e^{-kt_1}=\frac\alpha2,\qquad \boxed{t_1=\frac1k\log\frac{2(\alpha-1)}{\alpha-2}.}
$$
Since $\alpha>2$, 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 $t_1$:
$$
\boxed{T(t)=T_0+\left[\left(\frac\alpha2-1\right)T_0+\beta\right]e^{-k(t-t_1)},\qquad t>t_1.}
$$
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.
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