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.