Problem 30 - Thomas Calculus

Question

In Exercises \(25-30\) , obtain a slope field and add to it graphs of the solution curves passing through the given points. $$ \begin{array}{l}{y^{\prime}=\frac{x y}{x^{2}+4} \text { with }} \\\ {\begin{array}{lll}{\text { a. }(0,2)} & {\text { b. }(0,-6)} & {\text { c. }(-2 \sqrt{3},-4)}\end{array}}\end{array} $$

Step-by-Step Solution

Answer

Plot the slope field for \( y' = \frac{xy}{x^2 + 4} \) and draw solution curves through each given point.

Step 1: Understand the Differential Equation

The given differential equation is \( y' = \frac{xy}{x^2 + 4} \). This is a first-order ordinary differential equation which specifies how the slope \( y' \), or the derivative of \( y \) with respect to \( x \), behaves at each point \((x, y)\) in the plane.

Step 2: Plot the Slope Field

To plot the slope field, evaluate the expression \( \frac{xy}{x^2 + 4} \) at a grid of points. Each point \((x, y)\) on the plane corresponds to a slope \( m = \frac{xy}{x^2 + 4} \). Draw short line segments with these slopes at each grid point to visualize the behavior of the differential equation.

Step 3: Draw Solution Curves through Given Points

To find solution curves passing through the points \((0,2)\), \((0,-6)\), and \((-2\sqrt{3}, -4)\), integrate the slope field starting from each point. This involves following the direction indicated by the slope field while ensuring that the curve goes through the specified initial condition.

Step 4: Sketch the Full Solution Graph

Using the plotted slope field as a guide, sketch solution curves that smoothly follow the flow of the direction field, entering and exiting each of the given initial points. These curves represent specific solutions to the differential equation starting from the given points.

Key Concepts

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation involving a function and its derivatives. In this problem, the ODE given is \( y' = \frac{xy}{x^2 + 4} \). This equation describes how the derivative of \( y \), or the rate at which \( y \) changes with respect to \( x \), varies depending on the values of \( x \) and \( y \). ODEs are fundamental in describing many physical phenomena, such as motion, heat, or population growth.

• First-order ODEs involve only the first derivative of \( y \), as seen here with \( y' \).
• They often describe dynamic systems, where the state of the system changes over time or space.

Understanding an ODE means grasping how solutions can behave over different regions of the plane, and how these solutions are inherently tied to the initial conditions provided.

Solution Curves

Solution curves are graphical representations of all possible solutions that satisfy a differential equation, given certain initial conditions. In our example, these curves need to pass through the specific points \((0,2)\), \((0,-6)\), and \((-2\sqrt{3}, -4)\).To draw these curves:

• Start at the given point.
• Follow the slope direction indicated by the slope field closely.
• Continue drawing while ensuring the curve maintains the required initial condition.

The beauty of solution curves is they illustrate how the behavior of a differential equation evolves over the plane, providing a visual narrative of the dynamics at play. These curves help in spotting patterns and trends that are not immediately apparent from the equation alone.

Differential Equation Visualization

Visualizing a differential equation through a slope field is a powerful technique that bridges the gap between abstract equations and tangible graphical representations. A slope field, or direction field, is a grid of tiny line segments.

• Each segment represents the slope calculated from the differential equation at a particular point \((x, y)\).
• These line segments create a visible pattern on the plane, revealing how solutions to the differential equation might flow.

To create this visualization, one must evaluate the slope \( \frac{xy}{x^2 + 4} \) at numerous points. The resulting grid of directions gives insight into the nature and tendencies of the solutions, serving as a blueprint to draw solution curves.

Slope at Grid Points

The slope at grid points is calculated by plugging coordinates \((x, y)\) into the expression of the ODE. For the current task, the slope is given by \( m = \frac{xy}{x^2 + 4} \).

• This calculation provides a multitude of instantaneous rates of change at various locations on the plane.
• Plotting these slopes as short line segments at corresponding grid points creates the slope field.

Understanding slope at grid points is crucial as it influences the drawing of the solution curves. Each miniature slope segment guides the trajectory of these curves, ensuring they align with the expected behavior dictated by the differential equation.