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Use Euler's method to approximate the solution of the equation subject to the given conditions. a) =x+y, y(0) = 1 Ax=0.1 (step size) n= 8 (number of steps) dy dx

Use Euler's method to approximate the solution of the equation subject to the given conditions. a) =x+y, y(0) = 1 Ax=0.1 (step size) n= 8 (number of steps) dy dx

Advanced Engineering Mathematics
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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**Using Euler's Method to Approximate Solutions** Euler’s method is a numerical technique used to approximate the solution of first-order differential equations. It is particularly useful when an analytical solution is difficult or impossible to obtain. Below, we apply Euler’s method to solve the given differential equation under specific conditions. **Given Differential Equation:** \[ \frac{dy}{dx} = x + y \] **Initial Condition:** \[ y(0) = 1 \] **Parameters for Euler's Method:** - **Step Size (\(\Delta x\)):** 0.1 - **Number of Steps (n):** 8 **Objective:** Use Euler’s method to approximate the solution to the differential equation over the specified number of steps and conditions. Euler's method will iterate through the steps, updating the value of \(y\) using the formula: \[ y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n) \] where \(f(x, y)\) is the expression representing \(\frac{dy}{dx}\). Here, \(f(x, y) = x + y\). --- By following these instructions, the approximate solution can be computed step by step, updating the \(x\) and \(y\) values throughout the iteration process.
Transcribed Image Text:**Using Euler's Method to Approximate Solutions** Euler’s method is a numerical technique used to approximate the solution of first-order differential equations. It is particularly useful when an analytical solution is difficult or impossible to obtain. Below, we apply Euler’s method to solve the given differential equation under specific conditions. **Given Differential Equation:** \[ \frac{dy}{dx} = x + y \] **Initial Condition:** \[ y(0) = 1 \] **Parameters for Euler's Method:** - **Step Size (\(\Delta x\)):** 0.1 - **Number of Steps (n):** 8 **Objective:** Use Euler’s method to approximate the solution to the differential equation over the specified number of steps and conditions. Euler's method will iterate through the steps, updating the value of \(y\) using the formula: \[ y_{n+1} = y_n + \Delta x \cdot f(x_n, y_n) \] where \(f(x, y)\) is the expression representing \(\frac{dy}{dx}\). Here, \(f(x, y) = x + y\). --- By following these instructions, the approximate solution can be computed step by step, updating the \(x\) and \(y\) values throughout the iteration process.
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Step 1: Introduction of the given problem

fraction numerator d y over denominator d x end fraction equals x plus y y open parentheses 0 close parentheses equals 1 increment x equals 0.1

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