Chapter 1: Q4E (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 $0.1 (h = 0.1)$ .
$\frac{d y}{d x} = \frac{x}{y}, y (0) = - 1$

Short Answer

Expert verified

$x_{n}$	0.1	0.2	0.3	0.4	0.5
$y_{n}$	-1	-1.01	-1.029	-1.085	-1.096

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

One has , $f (x, y) = \frac{- x}{y}, x_{0} = 0, y_{0} = - 1, h = 0.1$

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 y1

Now, find the value of $y_{1}$ when n = 0, then

$y_{1} = y_{0} + (0.1) (\frac{x_{0}}{y_{0}}) = - 1 + (0.1) (0) = - 1$

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

Put n=1 to find y2

The value of $y_{2}$ is

$y_{2} = y_{1} + (0.1) (\frac{x_{1}}{y_{1}}) = - 1 + (0.1) (\frac{0.1}{- 1}) = - 1.01$

Thus, the value of $y_{2} = - 1.01$ when $x_{2} = 0.2$

Put n=2 to find y3

Now the value of $y_{3}$ is

$y_{3} = y_{2} + (0.1) (\frac{x_{2}}{y_{2}}) = - 1.01 + (0.1) (\frac{0.2}{- 1.01}) = - 1.01 + (- 0.019) = - 1.029$

So, the value is $y_{3} = - 1.029$ when $x_{3} = 0.3$

Put n=3 to find y4

The value of $y_{4}$ is

$y_{4} = y_{3} + (0.1) (\frac{x_{3}}{y_{3}}) = - 1.029 + (0.1) (\frac{0.3}{- 1.029}) = - 1.029 + (- 0.029) = - 1.058$

Consequently, the value is $y_{4} = - 1.058$ when $x_{4} = 0.4$

Put n=4 to find y5

The value of $y_{5}$ is

$y_{5} = y_{4} + (0.1) (\frac{x_{4}}{y_{4}}) = - 1.058 + (0.1) (\frac{0.4}{- 1.058}) = - 1.058 + (- 0.0378) = - 1.096$

Therefore, the value is $y_{5} = - 1.096$ when $x_{5} = 0.5$

Therefore the solution is

$x_{n}$	0.1	0.2	0.3	0.4	0.5
$y_{n}$	-1	-1.01	-1.029	-1.058	-1.096

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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 $0.1 (h = 0.1)$ .
$\frac{d y}{d x} = \frac{x}{y}, y (0) = - 1$

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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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)=-1

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

Geometry

Decision Maths

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Applied Mathematics

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 $0.1 (h = 0.1)$ .
$\frac{d y}{d x} = \frac{x}{y}, y (0) = - 1$