Putting something with a different temperature in a space with a constant temperature. The assumption underlying here is that the overall room temperature stays constant (i.e. the thing that’s cooling is so small that it doesn’t hurt room temperature).
\begin{equation} y’(t) = -k(y-T_0) \end{equation}
where, \(T_0\) is the initial temperature.
The intuition of this modeling is that there is some \(T_0\), which as the temperature \(y\) of your object gets closer to t. The result we obtain
Solving
\begin{equation} \int \frac{\dd{y}}{y-T_0} = \int -k \dd{t} \end{equation}
we can solve this:
\begin{equation} \ln |y-T_0| = -kt+C \end{equation}
which means we end up with:
\begin{equation} |y-T_0| = e^{-kt+C} = e^{C}e^{-kt} \end{equation}
So therefore:
\begin{equation} y(t) = T_0 + C_1e^{-kt} \end{equation}
to include both \(\pm\) cases.
this tells us that cooling and heating is exponential. We will fit our initial conditions rom data to obtain \(C_1\).