A Governing Differential Equation for the Heat Loss of Coffee and Cream? Here's the problem statement: If you want your coffee hottest, do you add cream immediately or 30 minutes later? (Coffee = 90C and 12 oz., Cream = 15C and 1 oz.)
I'm fairly confident that the answer is to add the cream immediately. I have arguments based on the mechanisms of heat transfer, but I'm having trouble devising a governing differential equation related to the rate of heat loss and specific heat. This, I'm told, is essential for a solution. However, given that I've never taken differential equations I'm not really sure how to proceed... 
As I understand it, adding the cream would decrease the specific heat overall, but increase the rate of heat transfer due to the increased mass of the cream. I imagine that the equation should look something like this, but I'm really not sure: dQ/dt=cm(dT/dt)
Can anyone help me quantify this relation mathematically?
 A: 
I'm fairly confident that the answer is to add the cream immediately. 

This is an old classroom and examination problem and kind of a trick question as a class' opinion invariably breaks about 50/50, with regards to which method delivers the hottest coffee!
To solve it we make some simple assumptions, firstly that the cooling of the coffee, with or without immediate addition of the cream, follows Newton's cooling law:
$$\frac{dQ}{dt}=hA(T-T_0)$$
Where $Q$ is heat energy, $t$ is time, $h$ the heat transfer coefficient, $A$ the total surface area exposed to ambient air, $T$ the temperature of the coffee and $T_0$ the temperature of the ambient air.
In an infinitesimal time interval $dt$ the coffee loses heat acc.:
$$dQ=-mcdT$$
With $m$ the mass of the coffee and $c$ its specific heat capacity (the negative sign is needed because $dT<0$).
Substituting gives us:
$$-mc\frac{dT}{dt}=hA(T-T_0)$$
Integration between $T_1, t=0$ and $T_2, t$ gives us:
$$\ln\frac{T_2-T_0}{T_1-T_0}=-\frac{hA}{mc}t \tag1$$
From which $T_2$ can be calculated.
The second assumption is a heat balance. When we add the cream to the coffee the temperature becomes $T_3$ and:
$$(m+m_c)cT_3=mcT+m_ccT_c$$
Where $T_3$ is the temperature after addition of the cream,  $T$ is the temperature of the coffee before addition of the cream, $T_c$ the temperature of the cream, $m_c$ the mass of the cream and $c$ the specific heat capacity of both cream and coffee, so that $c$ drops out:
$$(m+m_c)T_3=mT+m_cT_c \tag2$$
This allows to calculate $T_3$.
To determine which path leads to the hottest coffee:


*

*Calculate end temperature by assuming cream was added at the end. Use $Eq.1$ first, then $Eq.2$.

*Calculate end temperature by assuming cream was added at the start. Use $Eq.2$ first, then $Eq.1$.

Note: this answer initially stated that adding cream at the start or end of the cooling period does not affect end-temperature. That was in error and the answer has been edited accordingly.

