Q5E Use Euler鈥檚 method with step s... [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: Q5E (page 1)

Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .

Short Answer

Expert verified

$x_{n}$	1.1	1.2	1.3	1.4	1.5
$y_{n}$	0.1	0.209	0.325	0.444	0.564

Step by step solution

Write the recursive formula

For the solution use the Euler鈥檚 formula $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n})$

Apply recursive formula

We have, $f (x, y) = x - y^{2}, 鈥� 鈥� x_{0} = 1, 鈥� 鈥� y_{0} = 0, 鈥� 鈥� h = 0.1$

Then, $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n}) = y_{n} + (0.1) (x - y^{2})$

Put n=0 to find y1

Now, find the value of $y_{1}$

$y_{1} = y_{0} + (0.1) (x_{0} - y_{0}^{2}) = 0 + (0.1) (1) = 0.1$

Hence, the value of $y_{1} = 0.1$ when $x_{1} = 1.1$

Put n=1 to find y2

The value of $y_{2}$ is

$y_{2} = y_{1} + (0.1) (x_{1} - y_{1}^{2}) = 0.1 + (0.1) (1.1 - 0.01) = 0.1 + 0.109 = 0.209$

Consequently, the value is $y_{2} = 0.209$ when $x_{2} = 1.2$

Put n=2 to find y3

Now the value of $y_{3}$

$y_{3} = y_{2} + (0.1) (x_{2} - y_{2}^{2}) = 0.209 + (0.1) (1.2 - 0.043) = 0.209 + 0.116 = 0.325$

So, the value of $y_{3} = 0.325$ when $x_{3} = 1.3$

Put n=3 to find y4

The value of $y_{4}$ is

$y_{4} = y_{3} + (0.1) (x_{3} - y_{3}^{2}) = 0.325 + (0.1) (1.3 - 0.106) = 0.325 + 0.119 = 0.444$

Thus, the value is $y_{4} = 0.444$ when $x_{4} = 1.4$

Put n=4 to find y5

The value of $y_{5}$ is

$y_{5} = y_{4} + (0.1) (x_{4} - y_{4}^{2}) = 0.444 + (0.1) (1.4 - 0.197) = 0.444 + 0.12 = 0.564$

Therefore, the value is $y_{5} = 0.564$ when $x_{5} = 1.5$

Therefore the solution is

role="math" localid="1663939345323" $x_{n}$	1.1	1.2	1.3	1.4	1.5
$y_{n}$	0.1	0.209	0.325	0.444	0.564

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91影视

Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .

Short Answer

Step by step solution

Write the recursive formula

Apply recursive formula

Put n=0 to find y1

Put n=1 to find y2

Put n=2 to find y3

Put n=3 to find y4

Put n=4 to find y5

One App. One Place for Learning.

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91影视

Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problemy'=x-y2, y (1) = 0 at the points x=1.1,1.2,1.3,1.4and1.5.

Short Answer

Step by step solution

Write the recursive formula

Apply recursive formula

Put n=0 to find y1

Put n=1 to find y2

Put n=2 to find y3

Put n=3 to find y4

Put n=4 to find y5

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Mechanics Maths

Pure Maths

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use Euler鈥檚 method with step size h = 0.1 to approximate the solution to the initial value problem
$y' = x - y^{2}$ , y (1) = 0 at the points $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .