Q15 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 13-16, write a differential equation that fits the physical description. The rate of change in the temperature T of coffee at time t is proportional to the difference between the temperature M of the air at time t and the temperature of the coffee at time t.

Step-by-Step Solution

Answer

$\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{k}\left(\mathrm{M}-\mathrm{T}\right)$ where k is the constant of proportionality.

Step 1: Analyzing the given statement

By analyzing the given statement,           

$\frac{\mathrm{dT}}{\mathrm{dt}}\propto \left(\mathrm{M}-\mathrm{T}\right)$ , where T is the temperature of coffee at time t and M is the air temperature at time t.

Step 2: Writing the differential equation

$\frac{\mathrm{dT}}{\mathrm{dt}}=\mathrm{k}\left(\mathrm{M}-\mathrm{T}\right)$, where k is the constant of proportionality.

Hence, $\frac{\mathbf{dT}}{\mathbf{dt}}\mathbf{=}\mathbf{k}\left(\mathbf{M}\mathbf{-}\mathbf{T}\right)$ is the required differential equation.