Chapter 1: Q1E (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 $0.1 (h = 0.1)$ .
$\frac{dy}{dx} = \frac{- x}{y}$ , $y (0) = 4$

Short Answer

Expert verified

$x_{n}$	0.1	0.2	0.3	0.4	0.5
$y_{n}$	4	3.998	3.992	3.985	3.975

Step by step solution

Write the recursive formula

One has, $f (x, y) = \frac{- x}{y}, x_{0} = 0, y_{0} = 4, h = 0.1$ .

Then $y_{n + 1} = y_{n} + h . f (x_{n}, y_{n}) = y_{n} + (0.1) (\frac{- x_{n}}{y_{n}})$

Put n=0 to find role="math" localid="1663928029230" y1

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

$y_{2} = y_{1} + (0.1) (\frac{- x_{1}}{y_{1}}) = 4 + (0.1) (\frac{- 0.1}{4}) = 4 - 0.0025 = 3.998$

Thus, the value is $y_{2} = 3.998$ when $x_{2} = 0.2$

Put n=2 to find y3

Now the value of $y_{3}$

$y_{3} = y_{2} + (0.1) (\frac{- x_{2}}{y_{2}}) = 3.9975 + (0.1) (\frac{- 0.2}{3.9975}) = 3.9975 - 0.005 = 3.992$

So, the values get $y_{3} = 3.992$ when $x_{3} = 0.3$

Put n=3 to find y4

The value of $y_{4}$ is

$y_{4} = y_{3} + (0.1) (\frac{- x_{3}}{y_{3}}) = 3.9925 + (0.1) (\frac{- 0.3}{3.9925}) = 3.9925 - 0.0075 = 3.985$

Consequently, the value of $y_{4} = 3.985$ when $x_{4} = 0.4$

Put n=4 to find y5

The value of $y_{5}$ is

$y_{5} = y_{4} + (0.1) (\frac{- x_{4}}{y_{4}}) = 3.9849 + (0.1) (\frac{- 0.4}{3.9849}) = 3.9849 - 0.0100 = 3.975$

Therefore, the value of $y_{5} = 3.975$ when $x_{5} = 0.5$ .

Therefore, the solution is

localid="1663929244756" $x_{n}$	0.1	0.3	0.4	0.5	0.6
localid="1663929266027" $y_{n}$	4	3.998	3.992	3.985	3.975

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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 $0.1 (h = 0.1)$ .
$\frac{dy}{dx} = \frac{- x}{y}$ , $y (0) = 4$

Short Answer

Step by step solution

Write the recursive formula

Put n=0 to find role="math" localid="1663928029230" y1

Put n=2 to find y3

Put n=3 to find y4

Put n=4 to find y5

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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 0.1h=0.1.dydx=-xy,y(0)=4

Short Answer

Step by step solution

Write the recursive formula

Put n=0 to find role="math" localid="1663928029230" y1

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

Pure Maths

Statistics

Theoretical and Mathematical Physics

Logic and Functions

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 $0.1 (h = 0.1)$ .
$\frac{dy}{dx} = \frac{- x}{y}$ , $y (0) = 4$