Consider the initial value problem given below.y' = -y(0) = 9Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 9(₁-1) 'Apply one iteration of Euler's method in order to write the next term yn + 1 in terms of xn. Yn, and h, as needed.Yn+1 =1Substitute in f(xn Yn) = -5Yn into the previous form.Yn+1 =Use this recursive relationship to write a term k steps in the future Yn+k in terms of Xn: Yn, h, and k, as needed.Yn+k=For the approximation in question, write the number of steps that are needed in terms of the step size h.k== is 9 (1-1/2)aThus, the approximation k= steps after y₁=

Answered: Consider the initial value problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Consider the initial value problem given below.
y' = -
y(0) = 9
Show that when Euler's method is used to approximate the solution to this initial value problem at x = 4, then the approximation with step size h is 9
(₁-1) '
Apply one iteration of Euler's method in order to write the next term yn + 1 in terms of xn. Yn, and h, as needed.
Yn+1 =
1
Substitute in f(xn Yn) = -5Yn into the previous form.
Yn+1 =
Use this recursive relationship to write a term k steps in the future Yn+k in terms of Xn: Yn, h, and k, as needed.
Yn+k=
For the approximation in question, write the number of steps that are needed in terms of the step size h.
k=
= is 9 (1-1/2)
a
Thus, the approximation k= steps after y₁=

Expert Solution

Step 1

It is given that $y' = - \frac{1}{5} y$ with $y (0) = 9$ .

The objective is to show that when Euler method is used to approximate the initial value problem at $x = 4$ , then the approximation with step size $h$ is $9 {(1 - \frac{h}{5})}^{\frac{4}{h}}$ .

The Euler method is a numerical method to obtain the approximate solution.

It starts with the known value $y (x_{0}) = y_{0}$ and computes the unknown values $y_{1}, y_{2}, \dots, y_{n}$ successively using the formula $y_{i + 1} = y_{i} + h f (x_{i}, y_{i})$ , where $y' = f (x, y)$ is the given differential equation.