Problem 24 - Calculus

Question

Problems \(13-26\) deal with logistic equations \(y^{\prime}=c y-b y^{2}\). In a change to \(y=r Y\) and \(t=s T\), how are the initial values \(y_{0}\) and \(y_{0}^{\prime}\) related to \(Y_{0}\) and \(Y_{0}^{\prime \prime}\) ?

Step-by-Step Solution

Answer

\( y_0 = rY_0 \) and \( y_0' = \frac{r}{s} Y_0' \).

Step 1: Understand the Transformation

We begin by recognizing that the problem involves a logistic differential equation expressed as \( y' = c y - b y^2 \). The transformation given is \( y = rY \) and \( t = sT \). This implies that both the dependent variable \(y\) and the independent variable \(t\) are being changed by constant factors \(r\) and \(s\), respectively.

Step 2: Express Derivatives in Terms of New Variables

Next, we need to express the derivative \(y'\) in terms of \(Y\) and \(T\). Using the chain rule, \( y' = \frac{dy}{dt} = \frac{d(rY)}{dsT} = r \frac{dY}{s dT} = \frac{r}{s} Y' \). Similarly, for higher order derivatives, we would have \( y'' = \frac{r}{s^2} Y'' \). But we are primarily focused on the first order derivative \(y'\).

Step 3: Relate Initial Values Based on Transformation

The initial condition for \(y\) is \(y_0\) at some initial time \(t_0\). Using \(y = rY\), this transforms to \(y_0 = rY_0\). Similarly, the initial derivative using \( y' = \frac{r}{s} Y' \) gives us \( y_0' = \frac{r}{s} Y_0' \).

Step 4: Conclude the Relationship

The transformed equations of the initial values are derived: \( y_0 = rY_0 \) relates \(y_0\) to \(Y_0\), and \( y_0' = \frac{r}{s} Y_0' \) relates \(y_0'\) to \(Y_0'\). These expressions describe how the original initial values are scaled due to the changes in the dependent and independent variables.

Key Concepts

Variable Transformation

When dealing with logistic equations, like the one given in the exercise, a **variable transformation** can often simplify the problem. Here, we transform variables from one form to another. This specific transformation is useful in equations like the logistic equation \( y' = c y - b y^2 \). The transformation provided is \( y = rY \) and \( t = sT \).

• **\( y = rY \)**: This means that the dependent variable \( y \) is expressed as a constant multiple \( r \) of a new variable \( Y \).
• **\( t = sT \)**: Similarly, the independent variable \( t \) becomes \( sT \), where \( s \) is a constant and \( T \) is a new variable.

By applying these transformations, the complexity of the equation can be reduced, allowing us to solve it more easily in terms of \( Y \) and \( T \). It also provides a clearer relationship between the new and old forms of the equation.

Initial Value Problem

An **initial value problem** refers to a differential equation that is solved given specific values or conditions at a starting point, commonly called the initial values. Here, the initial values \( y_0 \) and \( y_0' \) relate to the new transformed variables \( Y_0 \) and \( Y_0' \).

• For \( y_0 \), we have \( y_0 = rY_0 \). This transforms the starting (initial) value of the dependent variable through the multiplication by the transformation factor \( r \).
• For \( y_0' \), the transformation \( y_0' = \frac{r}{s} Y_0' \) applies. This factor considers both transformations of \( y \) and \( t \), scaling the initial derivative accordingly.

Understanding how these initial values transform is crucial in solving the initial value problem, as it defines the starting conditions of the system and impacts how the solution evolves over time.

First-Order Derivatives

The term **first-order derivatives** refers to the rate of change of a function with respect to its variable, which is critical in differential equations. Here, the first-order derivative of the equation is expressed as \( y' = \frac{dy}{dt} \). When we apply our variable transformations, the derivative also transforms:

• By the chain rule, \( y' = \frac{d(rY)}{dt} = \frac{r}{s} \cdot \frac{dY}{dT} = \frac{r}{s} Y' \).

This expression shows that the first-order derivative with respect to the new variable \( Y \) is scaled by the factor \( \frac{r}{s} \). This scaling is essential as it adjusts the derivative to maintain accuracy in modeling the logistic equation, reflecting how quickly or slowly the dependent variable changes over time relative to the transformed variables.