By Euler's method find y(1) ,given dy/dx =2(x+y) ,y(1)=0

To solve the given differential equation \( \frac{dy}{dx} = 2(x + y) \) using Euler's method, we will follow these steps:

### Step 1: Problem Setup
We are given:
- \( \frac{dy}{dx} = 2(x + y) \)
- Initial condition \( y(1) = 0 \)
- We need to find \( y(1) \), assuming small increments.

Let’s select a small step size, \( h \), and proceed with Euler's method. However, the problem says \( y(1) \), but initial \( x_0 = 1 \), and \( y_0 = 0 \), so it's likely asking for a step forward, say to \( x = 1.1 \) or \( x = 1.2 \). I will assume a small step size, \( h = 0.1 \), unless you specify otherwise.

### Step 2: Euler's Formula
Euler's method updates \( y \) as:
\[
y_{n+1} = y_n + h f(x_n, y_n)
\]
where \( f(x, y) = 2(x + y) \), \( x_n \) and \( y_n \) are the values at the \(n\)-th step, and \( h \) is the step size.

### Step 3: Perform the Iteration

#### Initial values:
- \( x_0 = 1 \)
- \( y_0 = 0 \)
- Step size \( h = 0.1 \)

#### Iteration 1:
Using the formula, the first approximation for \( y_1 \) at \( x_1 = 1.1 \) is:
\[
y_1 = y_0 + h \cdot f(x_0, y_0)
\]
\[
y_1 = 0 + 0.1 \cdot 2(1 + 0) = 0.1 \cdot 2 = 0.2
\]

#### Iteration 2:
Now for \( x_2 = 1.2 \), we use:
\[
y_2 = y_1 + h \cdot f(x_1, y_1)
\]
\[
y_2 = 0.2 + 0.1 \cdot 2(1.1 + 0.2) = 0.2 + 0.1 \cdot 2(1.3) = 0.2 + 0.26 = 0.46
\]

So, \( y(1.2) \approx 0.46 \).

Would you like to proceed further with more steps, or is there a specific value you're looking for? 

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Here are five questions for further exploration:
1. What is Euler's method used for in differential equations?
2. How does the step size \( h \) affect the accuracy of the Euler method?
3. What are some limitations of Euler's method compared to other numerical methods?
4. How would the result change if we used a smaller step size, like \( h = 0.05 \)?
5. How does the slope function \( f(x, y) = 2(x + y) \) influence the solution?

**Tip:** Reducing the step size improves accuracy in Euler's method, but it also increases the number of computations required.

By Euler's method find y(1), given dy/dx = 2(x + y), y(1) = 0

Learn how to apply Euler's Method to approximate y(1) for the differential equation dy/dx = 2(x + y) with the initial condition y(1) = 0. This step-by-step guide uses a step size of h = 0.1 to demonstrate numerical methods for solving differential equations. Suitable for advanced high school and college-level students.

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