Question

Slope Field In Exercises \(33-36,\) (a) write a differential equation for the statement, (b) match the differential equation with a possible slope field, and (c) verify your result by using a graphing utility to graph a slope field for the differential equation. [The slope fields are labeled (a), (b),(c), and (d).] To print an enlarged copy of the graph, go to MathGraphs.com. The rate of change of \(y\) with respect to \(x\) is proportional to \(y^{2}\)

Short answer

The differential equation matching the statement 'The rate of change of \(y\) with respect to \(x\) is proportional to \(y^{2}\)' is \(\frac{dy}{dx} = ky^{2}\), where \(k\) is the proportionality constant.

Step-by-step solution

Deriving the Differential Equation

Since the rate of change of \(y\) with respect to \(x\) is proportional to \(y^{2}\), the differential equation would be \(\frac{dy}{dx} = ky^{2}\), where \(k\) is the proportionality constant.

Matching the Differential Equation to a Slope Field

The slope field for the differential equation \(\frac{dy}{dx} = ky^{2}\) would show lines with slopes that increase as \( |y| \) increases, since the rate of change of \(y\) is proportional to \(y^{2}\). Therefore, compare the slope fields (a), (b), (c), and (d) and choose the one with this characteristic.

Verifying the Result Using a Graphing Utility

Use a graphing utility to graph a slope field for the differential equation \(\frac{dy}{dx} = ky^{2}\). The resulting graph should match the slope field chosen in the previous step.

Key concepts

Differential Equation

A differential equation is a mathematical expression that relates a function with its derivatives. In the context of our exercise, it describes how the rate at which a quantity changes is related to the value of this quantity. For our differential equation, we have \( \frac{dy}{dx} = ky^2 \), where \( y \) is a function of \( x \) and \( k \) is a constant. Here, \( \frac{dy}{dx} \) tells us how \( y \) changes as \( x \) changes, making it a powerful tool for modeling dynamic systems. It captures how one quantity influences another through rates of change.

Proportionality Constant

The proportionality constant, represented by \( k \) in our differential equation, is a crucial element. It dictates the strength and direction of the relationship between the change in \( y \) and the square of \( y \) itself.

• If \( k \) is positive, \( y \) increases more rapidly as it grows larger.
• If \( k \) is negative, smaller values of \( y \) lead to larger decreases.

This constant helps define the specific behavior of the system you are modeling, by scaling the rate of change. Without \( k \), the equation wouldn't be able to provide a precise model of the real-world situation you're interested in.

Rate of Change

The rate of change in a differential equation like \( \frac{dy}{dx} = ky^2 \) indicates how quickly or slowly \( y \) changes as \( x \) varies. Understanding this rate is foundational for predicting behavior over time under given conditions.

• A positive rate of change signifies that \( y \) increases as \( x \) increases.
• A negative rate of change implies that \( y \) decreases as \( x \) increases.
• The magnitude of the rate of change shows how drastic these increases or decreases are.

Such insights are vital for comprehension of phenomena, whether they're physical, biological, or simply mathematical.

Graphing Utility

A graphing utility is a computational tool that allows you to visualize mathematical equations, including differential equations. It can graph functions and their corresponding slope fields, providing a visual representation of how solutions will behave.

For the equation \( \frac{dy}{dx} = ky^2 \), using a graphing utility helps confirm that the slope field matches the expected pattern: lines that steepen as the magnitude of \( y \) increases. These diagrams help identify

• Patterns and trends in the slope field,
• The overall behavior of solutions to the differential equation,
• Any potential errors in previous steps.

Overall, a graphing utility is an invaluable resource for visual learning and verifying analytic results through graphical exploration.