Chapter 1: Q5E (page 1)

Use Euler’s 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’s formula $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n})$

Apply recursive formula

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

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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Use Euler’s 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

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Use Euler’s 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

Applied Mathematics

Mechanics Maths

Pure Maths

Statistics

Logic and Functions

Geometry

Study anywhere. Anytime. Across all devices.

Use Euler’s 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 $x = 1.1, 1.2, 1.3, 1.4 and 1.5$ .