Question

Draw a direction field for the given differential equation and state whether you think that the solutions are converging or diverging. $$ y^{\prime}=\left(y^{2}+2 t y\right) /\left(3+t^{2}\right) $$

Short answer

Question: Analyze the direction field for the given differential equation: $$ y^{\prime}=\left(y^{2}+2 t y\right) /\left(3+t^{2}\right) $$ and determine whether the solutions are converging or diverging.

Step-by-step solution

Understand the given differential equation

The given differential equation is: $$ y^{\prime}=\left(y^{2}+2 t y\right) /\left(3+t^{2}\right) $$ It is a first order differential equation in terms of two variables \(y\) and \(t\), where \(y^{\prime}\) represents the derivative of \(y\) with respect to \(t\).

Create a table for the given equation in different variables

To create a direction field, we need to find the slope of the tangent \(m\) at different points along the \(y\) and \(t\) plane. We can do this by plugging different values of \(t\) and \(y\) into the given equation. Let's create a table to store these values and their corresponding slopes: $$ \begin{array}{c|c|c} \textbf{t} & \textbf{y} & \boldsymbol{y^{\prime}} \\ \hline -2 & -2 & \frac{(-2)^2 + 2(-2)(-2)}{3+(-2)^2} \\ -2 & 0 & \frac{0^2 + 2(-2)(0)}{3+(-2)^2} \\ -2 & 2 & \frac{(2)^2 + 2(-2)(2)}{3+(-2)^2} \\ 0 & -2 & \frac{(-2)^2 + 2(0)(-2)}{3+(0)^2} \\ 0 & 0 & \frac{0^2 + 2(0)(0)}{3+(0)^2} \\ 0 & 2 & \frac{(2)^2 + 2(0)(2)}{3+(0)^2} \\ 2 & -2 & \frac{(-2)^2 + 2(2)(-2)}{3+(2)^2} \\ 2 & 0 & \frac{0^2 + 2(2)(0)}{3+(2)^2} \\ 2 & 2 & \frac{(2)^2 + 2(2)(2)}{3+(2)^2} \\ \end{array} $$

Draw the direction field

Using the table created in Step 2, we can now draw the direction field by plotting the tangent lines at each point with the corresponding slope. The figure should give you a visualization of the solution curves of the differential equation.

Analyzing the direction field for convergence or divergence

By looking at the direction field, analyze the patterns and behavior of the solution curves. If the curves appear to be getting closer to each other (converging) or farther away from each other (diverging), make a statement about the convergence or divergence of the solutions. In conclusion, by following the steps above, we can draw the direction field for the given differential equation and determine whether the solutions are converging or diverging.

Key concepts

Understanding First Order Differential Equations

First order differential equations are fundamental in mathematics, often serving as the starting point in the study of differential equations. These equations involve functions and their first derivatives. The general form of a first order differential equation is given by
\[ \frac{dy}{dt} = f(t, y) \]
The equation provided in the exercise,
\[ y' = \frac{y^2 + 2ty}{3 + t^2} \]
is an example of a non-linear first order differential equation since the function of \(y\) and \(t\) is non-linear. The goal when dealing with such equations is to determine the function \(y(t)\) that satisfies the equation for given initial conditions. This means finding how the value of \(y\) changes with respect to the independent variable \(t\). This particular type of equation does not have a straightforward analytical solution and thus prompts the use of direction fields to visualize the behavior of its solutions across different values of \(t\) and \(y\).

• They express a relationship between a function and its derivative.
• They can model a wide range of real-world phenomena.
• The solution to a first order differential equation is a function rather than a numerical value.

Exploring Slope Fields

Slope fields, also known as direction fields, are a graphical representation of a first order differential equation. They give us a way to visualize the behavior of solutions without actually solving the equation.
To create a slope field, we calculate the slopes at various points \((t, y)\) using the given differential equation. In our exercise, we calculate \(y'\) for different combinations of \(t\) and \(y\), and the results represent the slopes of the tangent lines to the solution curves at those points. These tangents are then drawn as short line segments, and the pattern they form is the slope field.
The beauty of slope fields lies in their ability to show us where the function increases, decreases, or remains constant. For example:

• If the slope field contains horizontal line segments, the function does not change along those points.
• If the line segments point upwards/rightwards, the function is increasing at those points.
• If the line segments point downwards/rightwards, the function is decreasing at those points.

In essence, a slope field provides a visual approximation of the family of solutions to the differential equation and can be especially useful when an equation is too complex to solve analytically.

Analyzing Solution Curves: Convergence and Divergence

When studying differential equations, we're often interested in the behavior of their solutions over the long term. Two important concepts in this context are convergence and divergence.
Convergence occurs when the solution curves of a differential equation come together as they are extended, indicating that they are approaching a common path or point. For instance, in a slope field, if the tangent lines seem to be getting closer to one another as we move along the independent variable, we would say the solutions are converging.
Divergence, on the other hand, means that the solution curves move apart from each other, indicating that they are following increasingly different paths.
In the given exercise, students are asked to observe the direction field they created and determine whether the pattern of the solution curves suggests convergence or divergence. However, remember that without a global perspective, it can be tricky to draw definitive conclusions just from a localized portion of the slope field. Predicting the behavior of solution curves in the long term often requires a deeper analysis or a broader view of the slope field.

• Convergence might indicate stability in the model being studied.
• Divergence often signals that small changes can lead to substantially different outcomes, which could illustrate sensitivity in a system.
• The points at which solution curves converge can hint at the existence of an equilibrium solution.