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

Let's solve the given initial value problem using Euler's method.

### Problem:
You are given the differential equation:
\[
\frac{dy}{dx} = 2(x + y)
\]
with the initial condition \( y(0) = 1 \), and you are asked to find \( y(1) \).

### Step-by-step Process:

1. **Formula for Euler's Method**:
   \[
   y_{n+1} = y_n + h \cdot f(x_n, y_n)
   \]
   where \( f(x_n, y_n) = 2(x_n + y_n) \) (from the given differential equation), and \( h \) is the step size.

2. **Initial Values**:
   - \( x_0 = 0 \)
   - \( y_0 = 1 \) (given as \( y(0) = 1 \))
   
   Let's divide the interval [0, 1] into equal steps. If we choose \( h = 0.1 \) (so that we take 10 steps from \( x = 0 \) to \( x = 1 \)), we can proceed with the calculations.

3. **Iterative Calculation**:
   We'll compute step by step using \( y_{n+1} = y_n + h \cdot f(x_n, y_n) \).

   1. \( x_0 = 0 \), \( y_0 = 1 \):
      \[
      y_1 = y_0 + 0.1 \cdot 2(0 + 1) = 1 + 0.1 \cdot 2 = 1 + 0.2 = 1.2
      \]
   
   2. \( x_1 = 0.1 \), \( y_1 = 1.2 \):
      \[
      y_2 = y_1 + 0.1 \cdot 2(0.1 + 1.2) = 1.2 + 0.1 \cdot 2(1.3) = 1.2 + 0.26 = 1.46
      \]
   
   3. \( x_2 = 0.2 \), \( y_2 = 1.46 \):
      \[
      y_3 = y_2 + 0.1 \cdot 2(0.2 + 1.46) = 1.46 + 0.1 \cdot 2(1.66) = 1.46 + 0.332 = 1.792
      \]

   4. \( x_3 = 0.3 \), \( y_3 = 1.792 \):
      \[
      y_4 = y_3 + 0.1 \cdot 2(0.3 + 1.792) = 1.792 + 0.1 \cdot 2(2.092) = 1.792 + 0.4184 = 2.2104
      \]

   5. \( x_4 = 0.4 \), \( y_4 = 2.2104 \):
      \[
      y_5 = y_4 + 0.1 \cdot 2(0.4 + 2.2104) = 2.2104 + 0.1 \cdot 2(2.6104) = 2.2104 + 0.52208 = 2.73248
      \]

   6. \( x_5 = 0.5 \), \( y_5 = 2.73248 \):
      \[
      y_6 = y_5 + 0.1 \cdot 2(0.5 + 2.73248) = 2.73248 + 0.1 \cdot 2(3.23248) = 2.73248 + 0.646496 = 3.378976
      \]

   7. \( x_6 = 0.6 \), \( y_6 = 3.378976 \):
      \[
      y_7 = y_6 + 0.1 \cdot 2(0.6 + 3.378976) = 3.378976 + 0.1 \cdot 2(3.978976) = 3.378976 + 0.7957952 = 4.1747712
      \]

   8. \( x_7 = 0.7 \), \( y_7 = 4.1747712 \):
      \[
      y_8 = y_7 + 0.1 \cdot 2(0.7 + 4.1747712) = 4.1747712 + 0.1 \cdot 2(4.8747712) = 4.1747712 + 0.97495424 = 5.14972544
      \]

   9. \( x_8 = 0.8 \), \( y_8 = 5.14972544 \):
      \[
      y_9 = y_8 + 0.1 \cdot 2(0.8 + 5.14972544) = 5.14972544 + 0.1 \cdot 2(5.94972544) = 5.14972544 + 1.189945088 = 6.339670528
      \]

   10. \( x_9 = 0.9 \), \( y_9 = 6.339670528 \):
       \[
       y_{10} = y_9 + 0.1 \cdot 2(0.9 + 6.339670528) = 6.339670528 + 0.1 \cdot 2(7.239670528) = 6.339670528 + 1.4479341056 = 7.7876046336
       \]

Thus, the approximate value of \( y(1) \) using Euler's method with a step size of 0.1 is \( \boxed{7.7876} \).

---

Would you like further clarification or details?

### Related Questions:
1. How does reducing the step size \( h \) affect the accuracy of Euler's method?
2. Can Euler's method be used for second-order differential equations?
3. What are the limitations of Euler's method?
4. How can you improve the accuracy of Euler's method without reducing the step size?
5. What is the difference between Euler's method and the Runge-Kutta method?

**Tip**: Euler's method is an example of a first-order numerical method, meaning its error decreases linearly with smaller step sizes.

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

Learn how to apply Euler's method to solve the differential equation dy/dx = 2(x + y) with the initial condition y(0) = 1. Step-by-step solution using a step size of 0.1, and explanation of key numerical methods concepts. Suitable for advanced high school or introductory college students.

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