= Newton cooling with an instantaneous temperature jump
{c}
Under <Newton's law of cooling> with fixed ambient temperature $T_0$ and $k>0$, an instantaneous jump $\beta$ at $t_*$ sets $T(t_*+)=T(t_*-)+\beta$. The subsequent temperature is $T_0+[T(t_*-)+\beta-T_0]e^{-k(t-t_*)}$. Equivalently, for an initial temperature $T_i$, the full solution is
$$
T(t)=T_0+(T_i-T_0)e^{-kt}+\beta H(t-t_*)e^{-k(t-t_*)}.
$$
This piecewise model has the <distributional derivative> equation $T'+k(T-T_0)=\beta\delta(t-t_*)$, with the <Dirac delta distribution> representing the added temperature.
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