/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Use Taylor's method of order two... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 6

Use Taylor's method of order two to approximate the solution for each of the following initial-value problems. a. \(\quad y^{\prime}=\frac{2-2 t y}{t^{2}+1}, \quad 0 \leq t \leq 1, \quad y(0)=1\), with \(h=0.1\) b. \(\quad y^{\prime}=\frac{y^{2}}{1+t}, \quad 1 \leq t \leq 2, \quad y(1)=-(\ln 2)^{-1}\), with \(h=0.1\) c. \(\quad y^{\prime}=\left(y^{2}+y\right) / t, \quad 1 \leq t \leq 3, \quad y(1)=-2\), with \(h=0.2\) d. \(\quad y^{\prime}=-t y+4 t / y, \quad 0 \leq t \leq 1, \quad y(0)=1\), with \(h=0.1\)

Short Answer

Expert verified
Apply the Taylor's method of order two on each initial-value problem separately by following the general formula of the method. Evaluate the approximated solution for each initial condition.

Step by step solution

01

Understanding the Taylor's method of order two

The Taylor's method of order two is used to approximate solutions of differential equations. The general formula for this method is \(y(t + h) = y(t) + h y'(t) + \frac{1}{2}h^{2} y''(t)\) where \(y'(t)\) is the first derivative and \(y''(t)\) is the second derivative of y with respect to t at the given point t.
02

Applying the Taylor's method on \(y'(t)=\frac{2-2 t y}{t^{2}+1}\)

First, find the second derivative y''(t) of the given function. After that, substitute \(t = 0\), \(y(0) = 1\), \(y'(t)\) and \(y''(t)\) into the Taylor's method formula. Finally, evaluate the value of \(y(t + h) = y(0.1)\) using \(h = 0.1\).
03

Applying the same method on subsequent functions

Repeat Step 2 with the other functions: \(y'(t)=\frac{y^{2}}{1+t}\), \(y'(t)=\left(y^{2}+y\right) / t\) and \(y'(t)=-t y+4 t / y\). Remember to use the corresponding initial conditions and \(h\) values.
04

Verifying the possible solutions

Perform verification of the solutions by plugging the approximated \(y(t + h)\) values back into the original differential equations. This is to ensure the left-hand side (LHS) equals to the right-hand side (RHS), then the solutions are correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Differential Equations
Differential equations are mathematical expressions that involve functions and their derivatives. They describe how these functions change over time or space.
In the context of Taylor's method, differential equations are used to design models through which we can predict behaviors and trends.
Taylor's method specifically helps in solving differential equations by providing an approximated solution step-by-step.
  • The equation usually involves a function (often denoted as \( y' \), the first derivative), which shows the rate of change of the original function \( y \).
  • For instance, in our exercise, we have numerous differential equations like \( y' = \frac{2-2ty}{t^2+1} \).
  • Solving these equations exactly is not always feasible, hence numerical methods such as Taylor's method offer practical solutions.
Understanding the structure of these equations is key. Each equation represents a unique problem with its own initial conditions. Expert knowledge of how the derivatives relate to one another is crucial in navigating their solutions.
Initial-Value Problems
Initial-value problems are a class of differential equations. They establish the condition of the solution at the starting point, known as \( t_0 \).
This initial condition is critical because it influences how the solution unfolds at every subsequent point.
  • In essence, these problems look to solve differential equations by predicting future values based on some initial data.
  • The initial condition is typically specified in the form \( y(t_0) = y_0 \).
  • For example, in part (a) of the exercise, \( y(0) = 1 \) serves as the starting point for predictions.
The accuracy of predicting using Taylor's method heavily relies on these initial conditions. They set the stage for how the subsequent calculations will span out across the defined interval.
Second Derivative
The second derivative, denoted as \( y''(t) \), measures the rate at which the first derivative changes.
It provides insight into the curvature of the original function \( y(t) \), showing whether the function is concave or convex at any given point.
  • In Taylor's method of order two, the second derivative becomes crucial in refining the approximation of the function over an interval.
  • In the application of Taylor's method, you calculate \( y''(t) \) based on the given differential equations.
  • A practical understanding of finding this second derivative helps solve complex initial-value problems more accurately.
This step in the method helps build on the accuracy of the solution beyond just considering the initial function \( y' \). The second derivative ensures the trajectory of the solution stays true to the nature of the original problem.
Numerical Approximation
Numerical approximation in the context of differential equations refers to methods used to find approximate solutions rather than exact ones.
This is especially useful when dealing with complex differential equations where exact solutions are difficult or impossible to find.
  • Taylor's method offers a straightforward approach to obtaining these approximations with improved precision by including higher-order derivatives like the second derivative.
  • By using increments \( h \), or step sizes, Taylor's method gradually builds up a solution over the interval.
  • Such incremental calculations help visualize the behavior of the original function over time.
In this exercise, for example, Step 3 involves repeating calculations across different initial-value problems using respective step sizes, giving students practical understanding of how numerical approximations evolve from the initial conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of the 20th century by A. J. Lotka and V. Volterra (see, for example, [Lo1], [Lo2], and [Vo]). Consider the problem of predicting the population of two species, one of which is a predator, whose population at time \(t\) is \(x_{2}(t)\), feeding on the other, which is the prey, whose population is \(x_{1}(t)\). We will assume that the prey always has an adequate food supply and that its birth rate at any time is proportional to the number of prey alive at that time; that is, birth rate (prey) is \(k_{1} x_{1}(t)\). The death rate of the prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) \(=k_{2} x_{1}(t) x_{2}(t)\). The birth rate of the predator, on the other hand, depends on its food supply, \(x_{1}(t)\), as well as on the number of predators available for reproduction purposes. For this reason, we assume that the birth rate (predator) is \(k_{3} x_{1}(t) x_{2}(t)\). The death rate of the predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) \(=k_{4} x_{2}(t)\). Since \(x_{1}^{\prime}(t)\) and \(x_{2}^{\prime}(t)\) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations $$ x_{1}^{\prime}(t)=k_{1} x_{1}(t)-k_{2} x_{1}(t) x_{2}(t) \quad \text { and } x_{2}^{\prime}(t)=k_{3} x_{1}(t) x_{2}(t)-k_{4} x_{2}(t) $$ Solve this system for \(0 \leq t \leq 4\), assuming that the initial population of the prey is 1000 and of the predators is 500 and that the constants are \(k_{1}=3, k_{2}=0.002, k_{3}=0.0006\), and \(k_{4}=0.5\). Sketch a graph of the solutions to this problem, plotting both populations with time, and describe the physical phenomena represented. Is there a stable solution to this population model? If so, for what values \(x_{1}\) and \(x_{2}\) is the solution stable?

Use all the Adams-Bashforth methods to approximate the solutions to the following initial-value problems. In each case use exact starting values, and compare the results to the actual values. a. \(\quad y^{\prime}=t e^{3 t}-2 y, \quad 0 \leq t \leq 1, \quad y(0)=0\), with \(h=0.2\); actual solution \(y(t)=\frac{1}{5} t e^{3 t}-\frac{1}{25} e^{3 t}+\) \(\frac{1}{25} e^{-2 t}\) b. \(\quad y^{\prime}=1+(t-y)^{2}, \quad 2 \leq t \leq 3, \quad y(2)=1\), with \(h=0.2\); actual solution \(y(t)=t+\frac{1}{1-t}\). c. \(\quad y^{\prime}=1+y / t, \quad 1 \leq t \leq 2, \quad y(1)=2\), with \(h=0.2\); actual solution \(y(t)=t \ln t+2 t\). d. \(\quad y^{\prime}=\cos 2 t+\sin 3 t, \quad 0 \leq t \leq 1, \quad y(0)=1\), with \(h=0.2 ;\) actual solution \(y(t)=\) \(\frac{1}{2} \sin 2 t-\frac{1}{3} \cos 3 t+\frac{4}{3} .\)

Discuss consistency, stability, and convergence for the Implicit Trapezoidal method $$ w_{i+1}=w_{i}+\frac{h}{2}\left(f\left(t_{i+1}, w_{i+1}\right)+f\left(t_{i}, w_{i}\right)\right), \quad \text { for } i=0,1, \ldots, N-1, $$ with \(w_{0}=\alpha\) applied to the differential equation $$ y^{\prime}=f(t, y), \quad a \leq t \leq b, \quad y(a)=\alpha. $$

Use the Taylor method of order two with \(h=0.1\) to approximate the solution to $$ y^{\prime}=1+t \sin (t y), \quad 0 \leq t \leq 2, \quad y(0)=0. $$

In a book entitled Looking at History Through Mathematics, Rashevsky [Ra], pp. 103-110, considers a model for a problem involving the production of nonconformists in society. Suppose that a society has a population of \(x(t)\) individuals at time \(t\), in years, and that all nonconformists who mate with other nonconformists have offspring who are also nonconformists, while a fixed proportion \(r\) of all other offspring are also nonconformist. If the birth and death rates for all individuals are assumed to be the constants \(b\) and \(d\), respectively, and if conformists and nonconformists mate at random, the problem can be expressed by the differential equations $$ \frac{d x(t)}{d t}=(b-d) x(t) \quad \text { and } \quad \frac{d x_{n}(t)}{d t}=(b-d) x_{n}(t)+r b\left(x(t)-x_{n}(t)\right) $$ where \(x_{n}(t)\) denotes the number of nonconformists in the population at time \(t\). a. Suppose the variable \(p(t)=x_{n}(t) / x(t)\) is introduced to represent the proportion of nonconformists in the society at time \(t\). Show that these equations can be combined and simplified to the single differential equation $$ \frac{d p(t)}{d t}=r b(1-p(t)) $$ b. Assuming that \(p(0)=0.01, b=0.02, d=0.015\), and \(r=0.1\), approximate the solution \(p(t)\) from \(t=0\) to \(t=50\) when the step size is \(h=1\) year. c. Solve the differential equation for \(p(t)\) exactly, and compare your result in part (b) when \(t=50\) with the exact value at that time.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.