Chapter 1: Q2E (page 1)

In problems $1 - 4$ Use Euler’s 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, $y_{n + 1} = y_{n} + h \cdot f (x_{n}, y_{n}) = y_{n} + (0 . 1) (2 y_{n} {- y}_{n}^{2})$

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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In problems $1 - 4$ Use Euler’s 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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In problems 1-4Use Euler’s 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

Applied Mathematics

Logic and Functions

Discrete Mathematics

Pure Maths

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

In problems $1 - 4$ Use Euler’s 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).
$\frac{dy}{dx} = y (2 - y)$ , $y (0) = 3$