Q2E In problems 聽1-4聽Use Euler鈥檚... [FREE SOLUTION]

91影视

Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 1: Q2E (page 1)

In problems $1 - 4$ Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).
$\frac{dy}{dx} = y (2 - y)$ , $y (0) = 3$

Short Answer

Expert verified

$x_{n}$	0.1	0.2	0.3	0.4	0.5
$y_{n}$	2.7	2.511	2.383	2.291	2.225

Step by step solution

Writing the recursive formula

We have, $f (x, y) = y (2 - y), x_{0} = 0, y_{0} = 3, h = 0.1$

Then, $y_{n + 1} = y_{n} + h 路 f (x_{n}, y_{n}) = y_{n} + (0 . 1) (2 y_{n} {- y}_{n}^{2})$

Putting n=0 to find y1

$y_{1} = y_{0} + (0 . 1) (2 y_{1} {- y}_{1}^{2}) = 3 + (0 . 1) (6 - 9) = 3 + (- 0 . 3) = 2.7$

The value of $y_{1} = 2.7$ for $x_{1} = 0.1$

Putting n=1 to find y2

$y_{2} = y_{1} + (0 . 1) (2 y_{1} {- y}_{1}^{2}) = 2 . 7 + (0 . 1) (5 . 4 - 7 . 29) = 2 . 7 + (- 0 . 189) = 2.511$

The value of $y_{2} = 2.511$ for $x_{2} = 0.2$

Putting n=2 to find y3

$y_{3} = y_{2} + (0 . 1) (2 y_{2} {- y}_{2}^{2}) = 2 . 511 + (0 . 1) (5 . 022 - 6 . 305) = 2 . 511 + (- 0 . 128) = 2.383$

The value of $y_{3} = 2.383$ for $x_{3} = 0.3$

Putting n=3 to find y4

$y_{4} = y_{3} + (0 . 1) (2 y_{3} {- y}_{3}^{2}) = 2 . 383 + (0 . 1) (4 . 766 - 5 . 679) = 2 . 383 + (- 0 . 091) = 2.291$

The value of $y_{4} = 2.291$ for $x_{4} = 0.4$

Putting n=4 to find y5

$y_{5} = y_{4} + (0 . 1) (2 y_{4} {- y}_{4}^{2}) = 2 . 291 + (0 . 1) (4 . 582 - 5 . 249) = 2 . 291 + (- 0 . 066) = 2.225$

The value of $y_{5} = 2.225$ for $x_{5} = 0.5$

Hence, the solution is

$x_{n}$	0.1	0.2	0.3	0.4	0.5
$y_{n}$	2.7	2.511	2.383	2.291	2.225

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91影视

In problems $1 - 4$ Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).
$\frac{dy}{dx} = y (2 - y)$ , $y (0) = 3$

Short Answer

Step by step solution

Writing the recursive formula

Putting n=0 to find y1

Putting n=1 to find y2

Putting n=2 to find y3

Putting n=3 to find y4

Putting n=4 to find y5

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91影视

In problems 1-4Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).dydx=y(2-y),y(0)=3

Short Answer

Step by step solution

Writing the recursive formula

Putting n=0 to find y1

Putting n=1 to find y2

Putting n=2 to find y3

Putting n=3 to find y4

Putting n=4 to find y5

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Geometry

Probability and Statistics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In problems $1 - 4$ Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).
$\frac{dy}{dx} = y (2 - y)$ , $y (0) = 3$