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.