Ein K?rper besitze zur Zeit \(t=0\) die Temperatur \(T_{0}\) und werde in der Folgezeit durch vorbeiströmende Luft der konstanten Temperatur \(T_{L}\) gekühlt \(\left(T_{L}0) $$ beschrieben (a: Konstante), Bestimmen Sie den zeitlichen Verlauf der K?rpertemperatur \(T\) für den Anfangswert \(T(0)=T_{0}\) durch Trennung der Variablen. Gegen welchen Endwert strebt die Körpertemperatur?

A differential equation is a mathematical equation that relates some function with its derivatives. In the context of this exercise, it is used to model how the temperature of a body changes over time. Differential equations can describe various phenomena in physics, engineering, biology, and other sciences.
In this exercise, the differential equation given is: \ \ \ \ \[ \ \ \frac{d T}{d t} = -a (T - T_{L})\].
This equation represents Newton’s Law of Cooling, where:

• \( T \) is the temperature of the body at time \( t \)
• \( T_{L} \) is the temperature of the surrounding air
• \( a \) is a positive constant that determines the rate of cooling

The left side \( \frac{d T}{d t} \) indicates how the temperature changes over time, and the right side \( -a (T - T_{L}) \) ties that change to the difference between the body’s temperature and the air temperature.

Variable separation is a method used to solve differential equations. It involves rearranging the equation so that each variable and its differential are on opposite sides of the equation. This way, each side can be integrated separately.
Starting with our differential equation: \[ \ \frac{dT}{dt} = -a(T - T_{L}) \].
We rearrange it to separate the variables \( T \) and \( t \):
\[ \frac{dT}{\(T - T_{L}) \)} = -a dt \].
Now, the left side contains only \( T \) and \( dT \), while the right side contains only \( t \) and \( dt \). With the variables separated, we can integrate both sides independently.

This is the crucial step because it turns our differential equation into an integrable form, allowing us to solve for \( T \) in terms of \( t \).

An initial value problem involves solving a differential equation by finding a specific solution that satisfies an initial condition. In our case, the initial condition is the temperature at time \( t = 0 \), which is given as \( T(0) = T_{0} \).

After integrating the separated variables and solving for \( T \), we encounter a constant of integration, \( C \). To find this constant, we use the initial condition:
\[ T_{0} - T_{L} = C^{'} e^{0} \]
\[ C^{'} = T_{0} - T_{L} \].
Applying this to our general solution, we get the specific solution:
\( T(t) = T_{L} + (T_{0} - T_{L})e^{-at} \)
This solution satisfies the initial condition, making it unique to our problem.

The temperature cooling model based on Newton's Law of Cooling describes how the temperature of an object changes over time in response to the temperature of the surrounding environment.
Our final form of the solution is:
\( T(t) = T_{L} + (T_{0} - T_{L})e^{-at} \)
This equation indicates that as time \( t \) increases, the exponential term \( e^{-at} \) decreases, causing the temperature \( T(t) \) to approach the ambient temperature \( T_{L} \).

• When \( t \to \infty \), \( e^{-at} \to 0 \)
• The body temperature \( T(t) \) then approaches \( T_{L} \), the surrounding air temperature.

This model is applicable not just for cooling, but for any process where the rate of change is proportional to the difference between the current state and some equilibrium state.