Use Euler's method with step size 0.3 to estimate y(1.5), where y(x) is the solution of the initial-value problem y' = 3x + y, y(0) = 1. y(1.5)
Use Euler's method with step size 0.3 to estimate y(1.5), where y(x) is the solution of the initial-value problem y' = 3x + y, y(0) = 1. y(1.5)
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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![### Euler's Method for Estimating Values
Use Euler's method with step size 0.3 to estimate \( y(1.5) \), where \( y(x) \) is the solution of the initial-value problem:
\[ y' = 3x + y^2, \quad y(0) = 1. \]
\[ y(1.5) = \boxed{\quad} \]
---
In this problem, we are given an initial value problem for a differential equation. We need to use Euler's method with a step size of 0.3 to estimate the value of the function \( y(x) \) at \( x = 1.5 \). Euler's method is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is a first-order method based on the linear approximation of the function.
To use Euler's method, follow these steps:
1. **Identify the initial conditions**: \( y(0) = 1 \).
2. **Determine the step size**: \( h = 0.3 \).
3. **Calculate the number of steps needed to reach \( x = 1.5 \)**: Using \( h = 0.3 \), we divide the interval from 0 to 1.5 into equal steps.
4. **Iteratively apply Euler's formula**:
\[
y_{n + 1} = y_n + h \cdot f(x_n, y_n)
\]
---
This text can serve as an instructional guideline and example for students learning numerical methods, specifically Euler's method, for solving initial-value problems in differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F4507a022-107e-448a-ac90-d5b99c5a3d21%2F8qodyu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Euler's Method for Estimating Values
Use Euler's method with step size 0.3 to estimate \( y(1.5) \), where \( y(x) \) is the solution of the initial-value problem:
\[ y' = 3x + y^2, \quad y(0) = 1. \]
\[ y(1.5) = \boxed{\quad} \]
---
In this problem, we are given an initial value problem for a differential equation. We need to use Euler's method with a step size of 0.3 to estimate the value of the function \( y(x) \) at \( x = 1.5 \). Euler's method is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is a first-order method based on the linear approximation of the function.
To use Euler's method, follow these steps:
1. **Identify the initial conditions**: \( y(0) = 1 \).
2. **Determine the step size**: \( h = 0.3 \).
3. **Calculate the number of steps needed to reach \( x = 1.5 \)**: Using \( h = 0.3 \), we divide the interval from 0 to 1.5 into equal steps.
4. **Iteratively apply Euler's formula**:
\[
y_{n + 1} = y_n + h \cdot f(x_n, y_n)
\]
---
This text can serve as an instructional guideline and example for students learning numerical methods, specifically Euler's method, for solving initial-value problems in differential equations.
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