Question

Temperature At time \(t=0\) minutes, the temperature of an object is \(140^{\circ} \mathrm{F} .\) The temperature of the object is changing at the rate given by the differential equation $$\frac{d y}{d t}=-\frac{1}{2}(y-72)$$ (a) Use a graphing utility and Euler's Method to approximate the particular solutions of this differential equation at \(t=1,2,\) and \(3 .\) Use a step size of \(h=0.1 .\) (A graphing utility program for Euler's Method is available at the website college.hmco.com.) (b) Compare your results with the exact solution $$y=72+68 e^{-t / 2} .$$ (c) Repeat parts (a) and (b) using a step size of \(h=0.05\) Compare the results.

Short answer

The accurate solutions calculated using the equation \(y=72+68e^{-t/2}\) will differ from those calculated by applying Euler's method with a step size of 0.1 or 0.05 due to the approximation of Euler's method. A decrease in step size typically results in a closer approximation to the accurate result.

Step-by-step solution

Set up the initial condition

The temperature at \(t=0\) is stated to be \(140°F = 68°C\). This will be the initial value for \(y\). So, the start point is \((0, 68)\).

Use Euler's method with step size h=0.1

Use Euler's method, where the new \(y\) value is calculated by adding the product of the differential at the current point and the step size, to the current value of \(y\). This is repeated iteratively to get values at \(t=1\), \(t=2\) and \(t=3\). Note that each step consists of ten 0.1 increments. Be sure to substitute the given equation, \(dy/dt = -1/2(y-72)\), into Euler's method.

Calculate the exact solutions

To find the accurate solutions for comparison, substitute \(t=1\), \(t=2\), and \(t=3\) into the given equation \(y=72+68e^{-t/2}\) to find the exact values of \(y\) at these times.

Compare the results

Compare the values obtained from the Euler's method to the exact solutions to assess the accuracy and efficiency of the method.

Repeat with step size h=0.05

Repeat steps 2 to 4 with a smaller step size of 0.05. Comparing these results with those from \(h=0.1\) will show how the error changes with step size.

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are fundamental in expressing many natural phenomena, such as the laws of motion, electrical circuits, and, as we see in the exercise, temperature change.

In the given problem, the temperature evolution of an object is modeled by a first-order differential equation \( \frac{d y}{d t}=-\frac{1}{2}(y-72) \). Here, \( y \) represents the temperature at time \( t \), and the rate of change of temperature with respect to time is given by the equation. The solution to such an equation will provide us with a function \( y(t) \) that describes the temperature of the object at any time \( t \).

Understanding differential equations allows students to predict and comprehend dynamic changes in various scientific and engineering fields. The step by step solution begins by identifying the initial condition, which is essential as it serves as the starting point for finding a particular solution to the differential equation.

Numerical Approximation

Numerical approximation techniques are used to find approximate solutions to mathematical problems that may not be easily solvable analytically. These methods are especially helpful when dealing with differential equations that describe complex systems.

In our example, Euler's Method is a numerical approximation technique used to solve the differential equation step by step. The essence of Euler's Method is starting from the known initial condition and taking fixed-size steps along the curve of the solution, estimating the slope from the differential equation at each step. Specifically, the new \( y \) value at each step is estimated by adding \( h \) times the derivative estimate to the current \( y \) value; mathematically, this is represented as \( y_{new} = y_{current} + h \cdot \frac{dy}{dt} \big|_{current} \).

By choosing a step size (\( h \) value), we determine the granularity of the approximation - smaller \( h \) generally leads to more accurate solutions but requires more computational effort. The exercise advises on comparing the results of different step sizes to illustrate the trade-off between computational effort and accuracy.

Temperature Change

Temperature change, in the context of this exercise, is modeled by the rate at which the temperature of an object adjusts to the surrounding environment. Differential equations can capture this process by relating the temperature of an object at a given time with its rate of change.

The specific rate of temperature change given by \( \frac{d y}{d t}=-\frac{1}{2}(y-72) \) suggests that the temperature change is proportional to the difference between the object's temperature \( y \) and the ambient temperature, taken to be \( 72^\circ F \) in this problem. The negative sign indicates that the object's temperature is decreasing towards the ambient temperature.

Predicting temperature change is crucial in many areas such as meteorology, climate science, and engineering. In the solution process, we observe through iterative calculations with Euler's Method how the temperature approaches the ambient temperature over time, reflecting on the physical phenomena of thermal equilibration. The negative exponential exact solution \( y=72+68e^{-t/2} \) fittingly captures this gradual change, with the temperature gradually stabilizing as time increases.