I'm solving a differential equation problem set and I bumped into the following DE problem where I got few question marks:
The temperature of a body at time $t$ is $T(t)$ and the temperature of its surrounding environment is $T_{env}$. In a small change in time
$t$ the temperature change of the body $T(t)$ is proportional to the
change in the amount of time $t$ and to the to difference between the
temperature of the body $T(t)$ and the temperature of the environment
$T_{env}$. Using this information (A) solve the differential equation
for temperature $T(t)$, (B) Solve $T(t)$ when the initial conditions
are: $T(0) = 10\,^{\circ}\mathrm{C}$ and $T_{env} = 50\,^{\circ}\mathrm{C}$
Now I actually already have an answer for this, it is (by the Newton's law in temperature change):
$$\frac{dT}{dt} = k(T(t)-T_{env})$$
My question is about the form of the question itself. If you were to present this question to a person with no prior knowledge of Newton's law in temperature change, would the phrase:
In a small change in time $t$ the temperature change of the body
$T(t)$ is proportional to the change in the amount of time $t$ and to
the to difference between the temperature of the body $T(t)$ and the
temperature of the environment $T_{env}$
give all the necessary information for the problem solver to be able to deduce (without no prior knowledge) that the connection is:
$$dT = k*dt*(T(t)-T_{env})$$
The problem states that:
The temperature change of the body $T(t)$ is proportional to the change in the amount of time $t$ and to
the to difference between the temperature of the body $T(t)$ and the
temperature of the environment $T_{env}$
But it doesn't explicitly state that the connection is:
$$dT = k*dt*(T(t)-T_{env})$$
It doesn't say that:
The temperature change of the body $T(t)$ is proportional to the multiplication of the change in the amount of time $t$ and to
the to difference between the temperature of the body $T(t)$ and the
temperature of the environment $T_{env}$
Without saying this explicitly can someone explain why couldn't it as well be e.g.:
$$dT = k\left[dt + (T(t)-T_{env})\right]$$
Of course the correct answer is more intuitive, but if you were a newbie in the field, how would you recognize this connection? Or would the problem need to be phrased more carefully?
For example:
If I would follow the same style as the problem I gave above, I could give another problem by:
A variable $x$ is proportional to variables $a$, $b$ and $c$. Solve
$x$
Don't you think the connection should be told more explicitly to the reader if it is assumed that there is no prior knowledge of the context?
Thnx for any help 🙂
Best Answer
Your doubt is lecit: let us check what does it happen if we consider the case $dT=kdt+kT-k\tilde{T}$. A Physicist would probably start with the difference equation (let us put $k=1$ for simplicity)
$$\Delta T(t)=\Delta t+T(t)-\tilde{T},~~(*)$$
for all $t\geq 0$ and check if its solutions are somehow compatible with the physical reality, where $\Delta T(t)=T(t)-T(t_1)$ and $\Delta t=t-t_1$. Choosing for example $t_1=0$, i.e. the starting moment of our analysis of the physical system, we obtain
$$T(t)-T(0)=t+T(t)-\tilde{T},$$
which leads to the "nonsense" $t=T(0)-\tilde{T}$. We can then deduce that the difference equation $(*)$ leads to no sound theory of the evolution of $T(t)$.
We are left to the interpretation $dT=kdt(T-\tilde{T})$ of the text in the OP. Even in this case we have to compare the analytical results with the physical reality: after all, we did not specify the sign of the constant $k\neq 0$...