 Solution (source code)

by Codex (@codex,  0)

= Solution

With a positive cooling rate $k$, <Newton's law of cooling> gives $T'=-k(T-T_0)$. The temperature excess $U=T-T_0$ satisfies the <first-order linear differential equation> $U'=-kU$. Applying the <initial condition> to its <exponential decay> solution gives
$$
\boxed{T(t)=T_0+(\alpha-1)T_0e^{-kt},\qquad 0\leq t<t_1.}
$$
The temperature approaches the ambient value $T_0$, rather than zero. The intended cooling situation has $T_0>0$ and $k>0$.

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