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Chapter 9: Problem 8

For the systems of differential equations in Exercises \(7-12\), use Euler's method with \(\Delta t=2\) to a) Plot the graphs of \(x\) and \(y\) for \(0 \leq t \leq 500\). b) Plot the trajectory of \(x\) and \(y\). \(x^{\prime}=x(0.1-0.01 x-0.004 y)\) \(y^{\prime}=y(0.05-0.001 x-0.0016 y), x(0)=20\) \(y(0)=4\)

Short Answer

Expert verified
Use Euler's method with \(\Delta t = 2\) to update \(x\) and \(y\) iteratively. Plot \(x(t)\) and \(y(t)\) as functions of time, and plot their trajectory in the phase plane.

Step by step solution

01

- Understand the Differential Equations

Given the system of differential equations: \[ x^{\prime}=x(0.1-0.01 x-0.004 y) \] \[ y^{\prime}=y(0.05-0.001 x-0.0016 y) \] with the initial conditions \( x(0)=20 \) and \( y(0)=4 \).
02

- Setup Euler's Method

Euler's method updates the values of \(x\) and \(y\) over increments of time \(\Delta t\). Here, the step size \(\Delta t = 2\). The general formulae for Euler's method are given by: \[ x_{n+1} = x_n + \Delta t \cdot x^{\prime}(t_n, x_n, y_n) \]\[ y_{n+1} = y_n + \Delta t \cdot y^{\prime}(t_n, x_n, y_n) \]
03

- Compute the First Few Steps

Start with the initial conditions \( x(0)=20 \), \( y(0)=4 \), and \( t_0 = 0 \). Compute the values for the first few steps using Euler's method: At \( t_0 = 0 \): \[ x^{\prime}(0) = 20(0.1 - 0.01 \cdot 20 - 0.004 \cdot 4) = 20(0.1 - 0.2 - 0.016) = 20(-0.116) = -2.32 \]\[ y^{\prime}(0) = 4(0.05 - 0.001 \cdot 20 - 0.0016 \cdot 4) = 4(0.05 - 0.02 - 0.0064) = 4(0.0236) = 0.0944 \]Updating the values:\[ x(2) = 20 + 2 \cdot (-2.32) = 15.36 \]\[ y(2) = 4 + 2 \cdot 0.0944 = 4.1888 \]
04

- Continue Computation

Continue using the same procedure for each subsequent step \(t_i\) for \(0 \leq t \leq 500\). This will generate the values for \(x(t)\) and \(y(t)\) at each time step.
05

- Plot the Graphs of x(t) and y(t)

Using the computed values, plot \(x(t)\) and \(y(t)\) as functions of time from \( t = 0 \) to \( t = 500 \). Tools such as Python's Matplotlib, Excel, or graphing calculators can be used.
06

- Plot the Trajectory in the Phase Plane

Create a phase plot with \(x(t)\) on the x-axis and \(y(t)\) on the y-axis using the same computed values. This plot shows the trajectory of the system in the \(xy\)-plane.

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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 equations that involve an unknown function and its derivatives. In this exercise, we're dealing with a system of first-order differential equations. These are equations where the highest derivative is the first derivative. The given equations are: \[ x^{prime} = x(0.1 - 0.01x - 0.004y) \] \[ y^{prime} = y(0.05 - 0.001x - 0.0016y) \]These equations describe how the functions \( x(t) \) and \( y(t) \) evolve over time. Differential equations are fundamental in many fields, such as physics, engineering, and biology. They help us understand how systems change dynamically.
Numerical Methods
Numerical methods are techniques to approximate the solutions of mathematical problems. Since many differential equations cannot be solved analytically (exactly), we rely on numerical methods like Euler's method. Euler's method approximates the solution of a differential equation by stepping forward in small increments from an initial value. In this exercise, we use the formula: \[ x_{n+1} = x_n + Delta t cdot x^{prime}(t_n, x_n, y_n) \] \[ y_{n+1} = y_n + Delta t cdot y^{prime}(t_n, x_n, y_n) \] Here, \( Delta t = 2 \). We start with initial conditions and compute new values step by step. This method helps us build a numerical solution over the interval \( 0 leq t leq 500 \).
Phase Plane Analysis
Phase plane analysis is a graphical technique to study systems of differential equations. It involves plotting the trajectory of the system in the \(xy\)-plane. In our exercise, we create a phase plot with \(x(t)\) on the x-axis and \(y(t)\) on the y-axis. This plot helps us visualize how the system behaves over time. The trajectory shows the combined evolution of \(x\) and \(y\), providing insights into the system's dynamics. For example, we can identify equilibrium points where the system is stable or observe any cyclic behavior.
Population Dynamics
Population dynamics is the study of how populations change over time. The given differential equations might represent a model in population biology, where \(x\) and \(y\) could be populations of two interacting species. For instance, they can represent prey and predator dynamics. The terms in the equations reflect growth rates and interaction effects: * The term \(0.1 - 0.01 x - 0.004 y\) in \( x^{prime} \) shows how \(x\) changes, including its natural growth and the effect of \(y\) on \(x\).* Similarly, \(0.05 - 0.001 x - 0.0016 y\) in \( y^{prime} \) indicates how \(y\) evolves.Understanding these effects helps us predict long-term population trends.
Graphing
Graphing is a crucial tool for visualizing data and functions. For this exercise, we create two types of graphs:* The graphs of \(x(t)\) and \(y(t)\) plot each function against time \(t\). These show how each variable changes individually over the interval \(0 \leq t \leq 500\).* The trajectory plot in the phase plane, showing \(x(t)\) versus \(y(t)\). This graph provides a combined view of their behaviors.Tools like Python鈥檚 Matplotlib, Excel, or graphing calculators can be used. Through graphing, complex mathematical results become more comprehensible.

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Most popular questions from this chapter

Let \(x\) and \(y\) represent the populations (in thousands) of two species that share a habitat. For each system of equations: a) Find the equilibrium points and assess their stability. Solve only for equilibrium points representing nonnegative populations. b) Give the biological interpretation of the asymptotically stable equilibrium point(s). \(x^{\prime}=x(0.1-0.01 x-0.005 y)\) \(y^{\prime}=y(0.05-0.001 x-0.002 y)\)

Zinc Depletion. After intake of zinc into the body, the zinc may be found in either the plasma or a portion of the liver. Let \(P(t)\) and \(L(t)\) be the amount of zinc in the plasma and liver after \(t\) days, respectively. The transfer rate of zinc from the plasma to the liver is 3 per day. The transfer rate from the liver to the plasma is \(0.6\) per day. Finally, the zinc is removed from the body via plasma at a rate of \(2.24\) per day. a) Draw a two-compartment model for \(L\) and \(P\). b) Find a system of differential equations satisfied by \(L\) and \(P\). c) Solve for \(P(t)\) and \(L(t)\) given that \(P(0)=0\) and \(L(0)=241 / 15\)

Tank A contains 100 gal of pure water. Tank B contains \(33 \mathrm{lb}\) of salt dissolved in 50 gal of water. Pure water is poured into tank \(B\) at the rate of \(3.5\) gal per min while an equal amount of the mixture is drained from the bottom of tank B. The mixture from tank \(A\) is pumped to tank \(B\) at the rate of 10 gal per min, while that from tank \(B\) is pumped to tank \(A\) at the same rate. Assume that the mixture in each tank is kept uniform by stirring. Let \(A(t)\) and \(B(t)\) be the amount of salt in tanks \(A\) and \(B\) after \(t\) minutes, respectively. a) Draw a two-compartment model for \(A(t)\) and \(B(t) .\) b) Show that \(A(t)\) and \(B(t)\) satisfy the differential equations \(A^{\prime}=-0.1 A+0.2 B\) and \(B^{\prime}=0.1 A-0.27 B\) c) Use the initial conditions \(A(0)=0\) and \(B(0)=33\) to solve for \(A\) and \(B\). d) Use a grapher to plot \(A(t)\) and \(B(t)\) for \(0 \leq t \leq 50\)

Solve the initial-value problem. \(x^{\prime}=2 x+3 y, y^{\prime}=-3 x+8 y, x(0)=1, y(0)=1\)

Find the general solution of the system of equations. \(x^{\prime}=5 x-y-5, y^{\prime}=2 x+2 y+2\)
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