Given dy/dx=y-x, y(0)=2, find y(0.1) using Runge Kutta second order method.

To solve the differential equation \( \frac{dy}{dx} = y - x \), with the initial condition \( y(0) = 2 \), using the second-order Runge-Kutta method (also called Heun’s method), we proceed as follows:

### Step-by-Step Approach

#### Step 1: Write down the function and initial values
- Differential equation: \( \frac{dy}{dx} = f(x, y) = y - x \)
- Initial condition: \( y(0) = 2 \)
- We want to find \( y(0.1) \), so let the step size be \( h = 0.1 \).

#### Step 2: Define the Runge-Kutta second-order formula
The second-order Runge-Kutta method (Heun's method) is given by the following iterative formula:

\[
y_{n+1} = y_n + \frac{h}{2} \left( k_1 + k_2 \right)
\]

where:
- \( k_1 = f(x_n, y_n) \)
- \( k_2 = f(x_n + h, y_n + h k_1) \)

#### Step 3: Perform the first step calculation

- Start with \( x_0 = 0 \), \( y_0 = 2 \), and \( h = 0.1 \).

**Calculate \( k_1 \):**

\[
k_1 = f(x_0, y_0) = y_0 - x_0 = 2 - 0 = 2
\]

**Calculate \( k_2 \):**

\[
k_2 = f(x_0 + h, y_0 + h k_1) = f(0 + 0.1, 2 + 0.1 \times 2) = f(0.1, 2.2)
\]
\[
k_2 = 2.2 - 0.1 = 2.1
\]

#### Step 4: Update \( y \)

Now use the Runge-Kutta formula to compute \( y_1 \) (which is \( y(0.1) \)):

\[
y_1 = y_0 + \frac{h}{2} (k_1 + k_2) = 2 + \frac{0.1}{2} (2 + 2.1)
\]
\[
y_1 = 2 + 0.05 \times 4.1 = 2 + 0.205 = 2.205
\]

### Conclusion
Thus, the approximate value of \( y(0.1) \) is \( 2.205 \).

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Would you like more details on this solution or have any further questions?

Here are five related questions that expand on this:

1. What is the difference between Runge-Kutta 2nd order and 4th order methods?
2. How do we improve the accuracy of the Runge-Kutta method?
3. Can you solve the same problem using Euler's method for comparison?
4. How does step size \( h \) affect the result in numerical methods?
5. What are the advantages of using higher-order methods in solving differential equations?

**Tip:** Always check how your numerical solution converges by reducing the step size and comparing results with exact or analytical solutions.

Learn how to solve the differential equation dy/dx = y - x using the second-order Runge-Kutta method. This detailed solution walks you through finding y(0.1) given y(0) = 2 with step-by-step calculations. Suitable for Grades 11-12 and college-level students.

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