Question

Direction fields with technology Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions \(y(0)=A\) lead to solutions that are increasing in time. $$y^{\prime}(t)=0.5(y+1)^{2}(t-1)^{2},|t| \leq 3 \text { and }|y| \leq 3$$

Short answer

Answer: The initial conditions that lead to increasing solutions are those where \(y(0)=A \neq -1\).

Step-by-step solution

Plot the direction field for the differential equation

Using a graphing utility, plot the direction field for \(y^{\prime}(t)=0.5(y+1)^{2}(t-1)^{2}\). Since there are many graphing utilities available, you should use the one you are most comfortable with. Desmos and Wolfram Alpha are popular graphing utilities, but you can use any one that can plot direction fields. For example, using Desmos, you can enter the differential equation as d(y)/d(x)=0.5(y+1)^2(x-1)^2 where x represents the independent variable t and y represents the dependent variable.

Find the constant solutions

A constant solution is a solution where \(y(t) = c\), where c is a constant, and \(y'(t) = 0\). We need to find the solutions that satisfy the condition. $$0 = 0.5(y+1)^{2}(t-1)^{2}$$ For this equation to be true, either \((y+1)^{2}\) or \((t-1)^{2}\) must be equal to zero. Let's first consider when \((y+1)^{2} = 0\). $$y+1 = 0$$ $$y = -1$$ So, one constant solution is \(y(t) = -1\). Now let's consider when \((t-1)^{2} = 0\). $$t-1 = 0$$ $$t = 1$$ However, this does not give a constant solution for y, so our only constant solution is \(y(t) = -1\).

Determine the initial conditions leading to increasing solutions

To find which initial conditions (\(y(0)=A\)) lead to increasing solutions, we should analyze the first derivative, \(y'(t)\). We are looking for the conditions where \(y'(t) > 0\). Using the given differential equation, we have: $$y'(t) = 0.5(y+1)^{2}(t-1)^{2}$$ We need to find the conditions for which this expression is greater than zero. We see that \((y+1)^{2}\) and \((t-1)^{2}\) are always non-negative (since they are squared), which means that their product is also non-negative. Therefore, \(y'(t)\) can only be zero or positive. We already found that a constant solution exists when \(y=-1\). To have an increasing solution, \(y'(t) > 0\), so \(y \neq -1\). This means that the initial conditions which lead to increasing solutions are those where \(A \neq -1\).

Key concepts

Graphing Utilities for Direction Fields

Understanding differential equations often involves visualizing their behavior, and this is where graphing utilities for direction fields become invaluable. Direction fields, also known as slope fields, provide a graphical representation of a differential equation by showing tangents to the hypothetical solution curves at specified points.

Students can use tools such as Desmos or Wolfram Alpha to input a given differential equation and instantly receive a plotted direction field. For example, in the exercise, the equation \(y^{\textprime}(t)=0.5(y+1)^{2}(t-1)^{2}\) is plotted for \(|t| \leq 3\) and \(|y| \leq 3\). In the graph, each line segment represents the slope of the solution curve at that point, giving insight into how the solution might behave.

These utilities often provide interactive elements, allowing students to adjust parameters and immediately see how the direction field changes. This level of interactivity is particularly useful for building intuition about the solutions—how they might converge, diverge, or follow along certain paths. Such tools are indispensable for students as they transform complex abstract equations into tangible visual information that can be more easily comprehended.

Constant Solutions of Differential Equations

Constant solutions in differential equations are particular solutions where the value of the dependent variable does not change over time. Identifying these solutions is an important aspect of solving differential equations as they reveal the steady states of the system being modeled.

In the given exercise, to find the constant solution for the equation \(y^{\textprime}(t)=0.5(y+1)^{2}(t-1)^{2}\), it's essential to set the derivative equal to zero since constant functions have a slope of zero. This process leads to discovering that \(y(t) = -1\) is the constant solution within the specified range.

Recognizing and understanding constant solutions can greatly simplify further analysis, as they often serve as reference points or baselines from which the behavior of other, more dynamic solutions can be compared. Constant solutions are particularly interesting in real-world applications, as they can represent equilibrium states in physical, biological, or economic systems.

Initial Conditions of Differential Equations

Initial conditions in the context of differential equations are essentially the starting points for our solutions—they anchor the equation to a specific scenario or state of the system. When dealing with a differential equation, the initial condition typically specifies the value of the dependent variable at a particular value of the independent variable. In our case, that's \(y(0)=A\).

The significance of initial conditions cannot be overstressed, as they can dramatically affect the trajectory of the solution over time. For our exercise, analyzing the initial conditions helps us determine which starting values \(A\) will lead to an increasing—or, technically speaking, a 'positive growth'—solution over time.

It's crucial for students to appreciate that the seemingly simple act of choosing an initial condition has profound implications on the resulting solution of a differential equation. By setting initial conditions, students learn to control the range of possible solutions and focus on the particular solution that corresponds to a real-world scenario or specific research question they are interested in. Initial conditions represent the bridge between the abstract world of mathematics and the concrete reality it aims to model.