Use Runge-Kutta method of second order to find y (0.2) .Given dy/dx=y-x,y(0) =2 ,h=0.1

To solve the differential equation \(\frac{dy}{dx} = y - x\) using the second-order Runge-Kutta method (also known as the midpoint method) to find \(y(0.2)\), with the initial condition \(y(0) = 2\) and step size \(h = 0.1\), we'll follow these steps:

### The Second-Order Runge-Kutta Method (Midpoint Method):
Given the differential equation \(\frac{dy}{dx} = f(x, y)\) and initial condition \(y(x_0) = y_0\), the update formula using the second-order Runge-Kutta method is:

\[
y_{n+1} = y_n + k_2
\]
where:
\[
k_1 = h \cdot f(x_n, y_n)
\]
\[
k_2 = h \cdot f\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right)
\]

### Step 1: Initialize the values
- \( x_0 = 0 \)
- \( y_0 = 2 \)
- \( h = 0.1 \)
- \( f(x, y) = y - x \)

### Step 2: Calculate \(y(0.1)\)
1. **Calculate \(k_1\)**:
   \[
   k_1 = h \cdot f(x_0, y_0) = 0.1 \cdot (2 - 0) = 0.1 \cdot 2 = 0.2
   \]

2. **Calculate \(k_2\)**:
   \[
   k_2 = h \cdot f\left(x_0 + \frac{h}{2}, y_0 + \frac{k_1}{2}\right) = 0.1 \cdot f\left(0 + 0.05, 2 + 0.1\right)
   \]
   \[
   f\left(0.05, 2.1\right) = 2.1 - 0.05 = 2.05
   \]
   \[
   k_2 = 0.1 \cdot 2.05 = 0.205
   \]

3. **Update \(y_1\) to find \(y(0.1)\)**:
   \[
   y_1 = y_0 + k_2 = 2 + 0.205 = 2.205
   \]

### Step 3: Calculate \(y(0.2)\)
1. **Update the values for the next step**:
   \[
   x_1 = 0.1, \quad y_1 = 2.205
   \]

2. **Calculate \(k_1\) for the next step**:
   \[
   k_1 = h \cdot f(x_1, y_1) = 0.1 \cdot (2.205 - 0.1) = 0.1 \cdot 2.105 = 0.2105
   \]

3. **Calculate \(k_2\) for the next step**:
   \[
   k_2 = h \cdot f\left(x_1 + \frac{h}{2}, y_1 + \frac{k_1}{2}\right) = 0.1 \cdot f\left(0.15, 2.205 + 0.10525\right)
   \]
   \[
   f\left(0.15, 2.31025\right) = 2.31025 - 0.15 = 2.16025
   \]
   \[
   k_2 = 0.1 \cdot 2.16025 = 0.216025
   \]

4. **Update \(y_2\) to find \(y(0.2)\)**:
   \[
   y_2 = y_1 + k_2 = 2.205 + 0.216025 = 2.421025
   \]

### Final Answer:
Thus, the value of \(y(0.2)\) is approximately \(2.421\).

Would you like further clarification on any of these steps?

### Related Questions
1. How does the second-order Runge-Kutta method compare to the fourth-order Runge-Kutta method?
2. Can you apply this method to a system of differential equations?
3. How would the solution change if the step size \(h\) were smaller?
4. What is the global error in the second-order Runge-Kutta method?
5. How can we use the Runge-Kutta method to approximate higher-order derivatives?

### Tip
When using numerical methods like Runge-Kutta, smaller step sizes \(h\) generally lead to more accurate results but increase computational effort.

Learn how to solve the differential equation dy/dx = y - x using the second-order Runge-Kutta method. Step-by-step guide with initial condition y(0) = 2 and step size h = 0.1. Detailed numerical approach and calculation for finding y(0.2).

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