# Solving differential equation regarding temperature change (Newton's law)

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 :)

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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 $k\neq 0$...

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+1 Thank you for your help :) – jjepsuomi Dec 4 '13 at 11:33
you are welcome! Try to solve the DE and analyse it w.r.t. the sign of $k$ and the difference $T(0)-\tilde{T}$ ;-) – Avitus Dec 4 '13 at 11:34
I will :) thnx ;D – jjepsuomi Dec 4 '13 at 11:35
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