### Euler's Method for Solving Initial-Value ProblemsThis section will guide you through using Euler's method to compute approximate solutions for an initial-value problem.#### Problem StatementWe 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 StepsTo 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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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

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

### 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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