Find an dy dx approximation of y(1) using Euler's Method and h=.2 for y(0)=1 = 4x-3y
Find an dy dx approximation of y(1) using Euler's Method and h=.2 for y(0)=1 = 4x-3y
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 Equation Approximation Using Euler's Method
**Problem Statement:**
Find an approximation of \( y(1) \) using Euler's Method with step size \( h = 0.2 \) for the differential equation:
\[
\frac{dy}{dx} = 4x - 3y
\]
with the initial condition:
\[
y(0) = 1
\]
### Explanation:
**Euler's Method:**
Euler's Method is an iterative approach to approximate solutions of first-order initial-value differential equations of the form \( \frac{dy}{dx} = f(x, y) \) with a given initial condition \( y(x_0) = y_0 \). The basic formula used in Euler's Method for a step size \( h \) is:
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
where \( y_{n+1} \) is the next value of \( y \), \( y_n \) is the current value of \( y \), and \( f(x_n, y_n) \) is the value of the differential equation evaluated at the current point \( (x_n, y_n) \).
Here, we are given:
\[
\frac{dy}{dx} = 4x - 3y \quad \text{and} \quad y(0) = 1
\]
We want to find \( y(1) \) using a step size \( h = 0.2 \).
### Steps of Calculation:
1. **Initialization:**
\[
x_0 = 0, \quad y_0 = 1, \quad h = 0.2
\]
2. **Iteration 1:**
\[
y_1 = y_0 + h (4x_0 - 3y_0)
\]
\[
y_1 = 1 + 0.2 (4(0) - 3(1))
\]
\[
y_1 = 1 - 0.6
\]
\[
y_1 = 0.4
\]
\[
x_1 = x_0 + h = 0 + 0.2 = 0.2
\]
3. **Iteration 2:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13675a72-9a5d-4ab9-ac06-e7f5015e8566%2F1185d7e5-60c4-48dc-8e48-ebc93c11a484%2F05iwz0f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equation Approximation Using Euler's Method
**Problem Statement:**
Find an approximation of \( y(1) \) using Euler's Method with step size \( h = 0.2 \) for the differential equation:
\[
\frac{dy}{dx} = 4x - 3y
\]
with the initial condition:
\[
y(0) = 1
\]
### Explanation:
**Euler's Method:**
Euler's Method is an iterative approach to approximate solutions of first-order initial-value differential equations of the form \( \frac{dy}{dx} = f(x, y) \) with a given initial condition \( y(x_0) = y_0 \). The basic formula used in Euler's Method for a step size \( h \) is:
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
where \( y_{n+1} \) is the next value of \( y \), \( y_n \) is the current value of \( y \), and \( f(x_n, y_n) \) is the value of the differential equation evaluated at the current point \( (x_n, y_n) \).
Here, we are given:
\[
\frac{dy}{dx} = 4x - 3y \quad \text{and} \quad y(0) = 1
\]
We want to find \( y(1) \) using a step size \( h = 0.2 \).
### Steps of Calculation:
1. **Initialization:**
\[
x_0 = 0, \quad y_0 = 1, \quad h = 0.2
\]
2. **Iteration 1:**
\[
y_1 = y_0 + h (4x_0 - 3y_0)
\]
\[
y_1 = 1 + 0.2 (4(0) - 3(1))
\]
\[
y_1 = 1 - 0.6
\]
\[
y_1 = 0.4
\]
\[
x_1 = x_0 + h = 0 + 0.2 = 0.2
\]
3. **Iteration 2:**
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