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Chapter 2: Problem 6

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

Short Answer

Expert verified
Explain your reasoning.

Step by step solution

01

Understand the given differential equation

The given differential equation is: $$ y^{\prime}=y(3-t y) $$ This is a first-order, nonlinear equation where the derivative of the dependent variable, y, with respect to the independent variable, t, depends on the product of y and \((3 - ty)\).
02

Draw a grid of points in the t-y plane

We will draw a grid of points in the t-y plane, with t on the horizontal axis and y on the vertical axis. Choose an appropriate range for t and y, such as, t going from -3 to 3 and y going from -3 to 3.
03

Find the slope at each grid point

With the grid points in place, we will determine the slope of the solution curve passing through each point. To do this, we'll use the equation for the derivative: $$ y^{\prime}=y(3-t y) $$ Find the slope by plugging in the grid point coordinates (t, y) and computing the value of the derivative (the slope) at that point.
04

Draw segments with these slopes at each grid point

Now that we have the slope at each grid point, we'll draw small line segments with these slopes at each grid point on the t-y plane. These short line segments will represent the direction field.
05

Examine the direction field to determine whether the solutions are converging or diverging

Observe the direction field drawn in the previous step. If the line segments appear to be converging toward a particular point or curve, we can conclude that the solutions of the given differential equation are converging. On the other hand, if the line segments appear to be diverging away from a certain point or curve, we can conclude that the solutions are diverging. Sometimes, the situation may not be clearly converging or diverging; in that case, we can describe the behavior of the solutions qualitatively or look for underlying patterns. In conclusion, follow these steps to analyze the given differential equation, draw its direction field, and determine whether the solutions are converging or diverging.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are a fundamental part of mathematics that describe how a particular quantity changes over time. In our exercise, the equation \( y' = y(3 - ty) \) represents a first-order, nonlinear differential equation. Here, \( y' \) is the derivative of \( y \) with respect to \( t \), which tells us the rate at which \( y \) changes as \( t \) changes. This equation is nonlinear due to the term \( ty \), which involves a product of variables. Analyzing such equations often involves finding a solution that explains how \( y \) behaves over time with initial conditions. Differential equations are widely used in fields like physics, engineering, and biology to model real-world phenomena, from population growth to the motion of objects.
Convergence and Divergence
In the context of differential equations, convergence and divergence describe the behavior of solution curves over time. A solution is said to be converging if it approaches a fixed value or curve as \( t \) increases. Conversely, a solution is diverging if it moves away from a point or a set trajectory. For the differential equation \( y' = y(3 - ty) \), observing a direction field can help us quickly see the likely behavior of solutions. If the line segments in the direction field move closer together and point to a common area, the solutions are converging. If they spread apart, the solutions are diverging. Understanding these concepts can help predict long-term behavior of dynamic systems.
Nonlinear Equations
Nonlinear equations, such as \( y' = y(3 - ty) \), involve terms that are not just simple multiples of the variables. The presence of the term \( ty \) in this equation makes it nonlinear. Nonlinear equations can exhibit complex behaviors such as limit cycles, chaos, or multiple solutions, depending on initial conditions and parameters. These equations are more challenging to solve analytically compared to linear equations and often require numerical methods for exploration. The nonlinearity can lead to unique patterns in the direction fields, reflecting the diverse possible trajectories of the solutions.
Slope Fields
Slope fields, also known as direction fields, are visual tools that help us understand differential equations without solving them explicitly. For an equation like \( y' = y(3 - ty) \), we use a grid in the \( t-y \) plane and compute the slopes given by the equation at various points. By plotting small line segments at each point with these slopes, we create a direction field. These line segments collectively demonstrate the behavior of the solutions to the differential equation. Slope fields can reveal whether solutions tend to converge to a stable value or diverge away from certain areas. This visual representation is a powerful method for gaining insight into complex equations without delving into intricate calculations.

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Most popular questions from this chapter

Harvesting a Renewable Resource. Suppose that the population \(y\) of a certain species of fish (for example, tuna or halibut) in a given area of the ocean is described by the logistic equation $$ d y / d t=r(1-y / K) y . $$ While it is desirable to utilize this source of food, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level, and possibly even driven to extinction. Problems 20 and 21 explore some of the questions involved in formulating a rational strategy for managing the fishery. In this problem we assume that fish are caught at a constant rate \(h\) independent of the size of the fish population. Then \(y\) satisfies $$ d y / d t=r(1-y / K) y-h $$ The assumption of a constant catch rate \(h\) may be reasonable when \(y\) is large, but becomes less so when \(y\) is small. (a) If \(hy_{0}>y_{1},\) then \(y \rightarrow y_{2}\) as \(t \rightarrow \infty,\) but that if \(y_{0}r K / 4,\) show that \(y\) decreases to zero as \(l\) increases regardless of the value of \(y_{0}\). (c) If \(h=r K / 4\), show that there is a single cquilibrium point \(y=K / 2\) and that this point is semistable (see Problem 7 ). Thus the maximum sustainable yield is \(h_{m}=r K / 4\) corresponding to the equilibrium value \(y=K / 2 .\) Observe that \(h_{m}\) has the same value as \(Y_{m}\) in Problem \(20(\mathrm{d})\). The fishery is considered to be overexploited if \(y\) is reduced to a level below \(K / 2\).

let \(\phi_{0}(t)=0\) and use the method of successive approximations to approximate the solution of the given initial value problem. (a) Calculate \(\phi_{1}(t), \ldots, \phi_{4}(t),\) or (if necessary) Taylor approximations to these iterates. Keep tems up to order six. (b) Plot the functions you found in part (a) and observe whether they appear to be converging. Let \(\phi_{n}(x)=x^{n}\) for \(0 \leq x \leq 1\) and show that $$ \lim _{n \rightarrow \infty} \phi_{n}(x)=\left\\{\begin{array}{ll}{0,} & {0 \leq x<1} \\ {1,} & {x=1}\end{array}\right. $$

Consider the initial value problem $$ y^{\prime}=3 t^{2} /\left(3 y^{2}-4\right), \quad y(1)=0 $$ (a) Use the Euler formula ( 6) with \(h=0.1\) to obtain approximate values of the solution at \(t=1.2,1.4,1.6,\) and 1.8 . (b) Repeat part (a) with \(h=0.05\). (c) Compare the results of parts (a) and (b). Note that they are reasonably close for \(t=1.2,\) \(1.4,\) and 1.6 but are quite different for \(t=1.8\). Also note (from the differential equation) that the line tangent to the solution is parallel to the \(y\) -axis when \(y=\pm 2 / \sqrt{3} \cong \pm 1.155 .\) Explain how this might cause such a difference in the calculated values.

Harvesting a Renewable Resource. Suppose that the population \(y\) of a certain species of fish (for example, tuna or halibut) in a given area of the ocean is described by the logistic equation $$ d y / d t=r(1-y / K) y . $$ While it is desirable to utilize this source of food, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level, and possibly even driven to extinction. Problems 20 and 21 explore some of the questions involved in formulating a rational strategy for managing the fishery. At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population \(y:\) The more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by \(E y,\) where \(E\) is a positive constant, with units of \(1 /\) time, that measures the total effort made to harvest the given species of fish. To include this effect, the logistic equation is replaced by $$ d y / d t=r(1-y / K) y-E y $$ This equation is known as the Schaefer model after the biologist, M. B. Schaefer, who applied it to fish populations. (a) Show that if \(E0 .\) (b) Show that \(y=y_{1}\) is unstable and \(y=y_{2}\) is asymptotically stable. (c) A sustainable yield \(Y\) of the fishery is a rate at which fish can be caught indefinitely. It is the product of the effort \(E\) and the asymptotically stable population \(y_{2} .\) Find \(Y\) as a function of the effort \(E ;\) the graph of this function is known as the yield-effort curve. (d) Determine \(E\) so as to maximize \(Y\) and thereby find the maximum sustainable yield \(Y_{m}\).

Consider the sequence \(\phi_{n}(x)=2 n x e^{-n x^{2}}, 0 \leq x \leq 1\) (a) Show that \(\lim _{n \rightarrow \infty} \phi_{n}(x)=0\) for \(0 \leq x \leq 1\); hence $$ \int_{0}^{1} \lim _{n \rightarrow \infty} \phi_{n}(x) d x=0 $$ (b) Show that \(\int_{0}^{1} 2 n x e^{-x x^{2}} d x=1-e^{-x}\); hence $$ \lim _{n \rightarrow \infty} \int_{0}^{1} \phi_{n}(x) d x=1 $$ Thus, in this example, $$ \lim _{n \rightarrow \infty} \int_{a}^{b} \phi_{n}(x) d x \neq \int_{a}^{b} \lim _{n \rightarrow \infty} \phi_{n}(x) d x $$ even though \(\lim _{n \rightarrow \infty} \phi_{n}(x)\) exists and is continuous.
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