 Newton cooling with an instantaneous temperature jump

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