Problem 34 - Thomas Calculus

Question

Exercises 33 and 34 have no explicit solution in terms of elementary functions. Use a CAS to explore graphically each of the differential equations. $$ \begin{array}{l}{\text { A Gompertz equation } y^{\prime}=y(1 / 2-\ln y), \quad y(0)=1 / 3} \\ {0 \leq x \leq 4, \quad 0 \leq y \leq 3 ; \quad y(3)}\end{array} $$

Step-by-Step Solution

Answer

Use a CAS to plot the solution curve and find \( y(3) \).

Step 1: Understanding the Gompertz Equation

The given differential equation is a Gompertz equation which is of the form \( y' = y \left( \frac{1}{2} - \ln y \right) \). The initial condition is \( y(0) = \frac{1}{3} \). This type of equation models growth processes where the growth rate decreases exponentially over time.

Step 2: Setup the Graphical Examination

To solve this equation graphically, input the equation \( y' = y \left( \frac{1}{2} - \ln y \right) \) into a Computer Algebra System (CAS). Set the initial condition \( y(0) = \frac{1}{3} \) and define the range for \( x \) from 0 to 4 and for \( y \) from 0 to 3.

Step 3: Analyze the Graph

Use the CAS to plot the solution curve of the differential equation within the specified domain. Observe the behavior of the curve with respect to the \( x \) and \( y \) axes to understand how the value of \( y \) changes over time (or as \( x \) increases) given the initial condition.

Step 4: Determine \( y(3) \)

Once the plot is generated, locate the point on the graph where \( x = 3 \). Read off the value of \( y \) at this coordinate. This is \( y(3) \), the value of the function at \( x = 3 \).

Key Concepts

Differential Equations

Differential equations are a type of equation that involve the rates of change of a quantity. They are fundamental to the description of dynamic systems. In simple terms, these equations describe how a particular quantity changes over time or another variable.
The Gompertz equation given above is a differential equation used to model growth processes. It is quite special because it involves an exponential decay in the rate of change.
In the Gompertz differential equation we've seen, the rate of change of the function, represented as \( y' \), depends on the current value of \( y \) and its natural logarithm, \(\ln y \).

• This specific form \[ y' = y \left( \frac{1}{2} - \ln y \right) \] captures the essence of growth rates that slow down as the system grows.

Understanding how to read and interpret such an equation is crucial for grasping how dynamic models work.

Initial Conditions

Initial conditions in differential equations are specific values at the starting point of the problem. They help us find the particular solution that fits the conditions we're interested in.
For example, in the Gompertz equation given, the initial condition is \( y(0) = \frac{1}{3} \). This tells us the starting value of the function \( y \) when \( x = 0 \).

• The purpose of specifying this initial condition is to fix the exact path the solution takes, among all the possible paths it could follow based on the differential equation.

Initial conditions are crucial because they help determine the unique solution to the differential equation. Without it, we would have an infinite set of potential solutions.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that helps solve mathematical equations symbolically and graphically. It provides a way to explore equations that might be challenging to solve analytically.
When dealing with our Gompertz equation, using a CAS allows us to input the equation along with its initial condition and visualize solutions over a specific range.

• With the CAS, you input the differential equation \[ y' = y \left( \frac{1}{2} - \ln y \right) \] and set parameters like the initial condition \( y(0) = \frac{1}{3} \), then analyze the solution from \( x = 0 \) to \( x = 4 \).
• It helps to observe the behavior and trends of the equation graphically, which can be more intuitive than trying to find a solution manually.

This type of graphical analysis can be crucial when an equation has no explicit solution, as it can help us understand how the system behaves visually.