se Euler's method with step size 0.5 to compute the approximate y-values Y1, Y2, Y3, and y4 of the olution of the initial-value problem -' = 1 + 3x + 2y, y(0) = 0. 1 = 2 3 = 4= Check Answer I| ||
se Euler's method with step size 0.5 to compute the approximate y-values Y1, Y2, Y3, and y4 of the olution of the initial-value problem -' = 1 + 3x + 2y, y(0) = 0. 1 = 2 3 = 4= Check Answer I| ||
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Euler's Method for Solving Initial-Value Problems
This section will guide you through using Euler's method to compute approximate solutions for an initial-value problem.
#### Problem Statement
We are given the differential equation:
\[ y' = 1 + 3x + 2y \]
with the initial condition:
\[ y(0) = 0 \]
Using Euler's method with a step size \( h = 0.5 \), we need to compute the approximate \( y \)-values \( y_1, y_2, y_3 \), and \( y_4 \).
Euler's method formula can be written as:
\[ y_{n+1} = y_n + h f(x_n, y_n) \]
where \( y' = f(x, y) \). In our case, \( f(x, y) = 1 + 3x + 2y \).
#### Computation Steps
To compute the desired values, we will follow these steps:
1. **Initial Step:**
\[
y_0 = y(0) = 0
\]
Plugging in \( x_0 = 0 \) and \( y_0 = 0 \) into the differential equation, we get:
\[
y' = 1 + 3(0) + 2(0) = 1
\]
Using Euler's formula for the next step:
\[
y_1 = y_0 + hf(x_0, y_0) = 0 + 0.5 \times 1 = 0.5
\]
2. **First Step:**
\[
x_1 = 0.5, \quad y_1 = 0.5
\]
Plugging in \( x_1 = 0.5 \) and \( y_1 = 0.5 \) into the differential equation, we get:
\[
y' = 1 + 3(0.5) + 2(0.5) = 3
\]
Using Euler's formula for the next step:
\[
y_2 = y_1 + hf(x_1, y_1) = 0.5 + 0.5 \times 3 = 2
\]
3. **Second Step](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc700bc18-61e4-4e04-a31d-52f10b21c2d0%2Fdccf79bd-8159-4953-991c-c31f1e93f8b4%2F948gv0m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Euler's Method for Solving Initial-Value Problems
This section will guide you through using Euler's method to compute approximate solutions for an initial-value problem.
#### Problem Statement
We are given the differential equation:
\[ y' = 1 + 3x + 2y \]
with the initial condition:
\[ y(0) = 0 \]
Using Euler's method with a step size \( h = 0.5 \), we need to compute the approximate \( y \)-values \( y_1, y_2, y_3 \), and \( y_4 \).
Euler's method formula can be written as:
\[ y_{n+1} = y_n + h f(x_n, y_n) \]
where \( y' = f(x, y) \). In our case, \( f(x, y) = 1 + 3x + 2y \).
#### Computation Steps
To compute the desired values, we will follow these steps:
1. **Initial Step:**
\[
y_0 = y(0) = 0
\]
Plugging in \( x_0 = 0 \) and \( y_0 = 0 \) into the differential equation, we get:
\[
y' = 1 + 3(0) + 2(0) = 1
\]
Using Euler's formula for the next step:
\[
y_1 = y_0 + hf(x_0, y_0) = 0 + 0.5 \times 1 = 0.5
\]
2. **First Step:**
\[
x_1 = 0.5, \quad y_1 = 0.5
\]
Plugging in \( x_1 = 0.5 \) and \( y_1 = 0.5 \) into the differential equation, we get:
\[
y' = 1 + 3(0.5) + 2(0.5) = 3
\]
Using Euler's formula for the next step:
\[
y_2 = y_1 + hf(x_1, y_1) = 0.5 + 0.5 \times 3 = 2
\]
3. **Second Step
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