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Calculus / Fundamentals of Differential Equations 8 / Chapter 3.7 / Problem 2E

Fundamentals of Differential Equations | 8th Edition | ISBN: 9780321747730 | Authors: R. Kent Nagle, Edward B. Saff, Arthur David Snider

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As in Exercises 3.6, for some problems you will find it

Chapter 3, Problem 2E

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QUESTION:

As in Exercises 3.6, for some problems you will find it essential to have a calculator or computer available.† For Problems 1–17, note whether or not \(\partial f / \partial y\) is bounded.

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem

\(y \prime=xy-y^2\), \(y(0)=-1\).

Equation Transcription:

$\partial{}f/\partial{}y$

$y'=xy-{y}^{2}$

$y(0)=-1$

Text Transcription:

partial f / partial y

y’=xy-y^2

y(0)=-1

Questions & Answers

QUESTION:

As in Exercises 3.6, for some problems you will find it essential to have a calculator or computer available.† For Problems 1–17, note whether or not \(\partial f / \partial y\) is bounded.

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem

\(y \prime=xy-y^2\), \(y(0)=-1\).

Equation Transcription:

$\partial{}f/\partial{}y$

$y'=xy-{y}^{2}$

$y(0)=-1$

Text Transcription:

partial f / partial y

y’=xy-y^2

y(0)=-1

ANSWER:

Solution:

Step 1

In this problem we have to determine the recursive formula for the Taylor method of order 2 for the initial value problem

$y'=xy-{y}^{2}$ , $y(0)=\ -1$

Recursive formula for the Taylor method of order $p$ is given by

${y}_{n+1}={y}_{n}+h\ f({x}_{n,}{y}_{n})+\frac{{h}^{2}}{2!}({f}_{2}({x}_{n,}{y}_{n})+................+\frac{{h}^{p}}{p!}({f}_{p}({x}_{n,}{y}_{n})$ .

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