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The problem provided involves using the Runge-Kutta fourth order method to find an approximate value of \( y \) when \( x = 0.1 \) for the differential equation given by:

\[
\frac{dy}{dx} = x^2 - y, \quad y(0) = 1
\]

Let's go through the Runge-Kutta fourth order (RK4) method step by step for this equation. RK4 is a numerical technique to solve ordinary differential equations and it provides a good balance between accuracy and computational cost.

### Step 1: Define the Problem
The differential equation is \( \frac{dy}{dx} = f(x, y) = x^2 - y \) with an initial condition \( y(0) = 1 \).

### Step 2: Set Parameters
We need to find the value of \( y \) at \( x = 0.1 \). Assume a step size \( h = 0.1 \) for simplicity (since we're taking just one step from \( x = 0 \) to \( x = 0.1 \)).

### Step 3: Runge-Kutta Fourth Order Method
The RK4 method calculates the next value \( y_{n+1} \) from \( y_n \) using the following formulas, where \( f(x, y) = x^2 - y \):

\[
k_1 = hf(x_n, y_n)
\]
\[
k_2 = hf(x_n + \frac{h}{2}, y_n + \frac{k_1}{2})
\]
\[
k_3 = hf(x_n + \frac{h}{2}, y_n + \frac{k_2}{2})
\]
\[
k_4 = hf(x_n + h, y_n + k_3)
\]
\[
y_{n+1} = y_n + \frac{k_1 + 2k_2 + 2k_3 + k_4}{6}
\]

### Step 4: Calculate using RK4

- \( x_0 = 0 \), \( y_0 = 1 \), \( h = 0.1 \)
- Calculate \( k_1, k_2, k_3, k_4 \) using the formulas above and \( f(x, y) = x^2 - y \)

Let's compute these values.The approximate value of \( y \) at \( x = 0.1 \) using the Runge-Kutta fourth order method, with the given initial condition and differential equation, is approximately \( 0.905 \).

Would you like any further details on this computation or have any other questions? Here are some additional topics we could explore:

1. How would changing the step size \( h \) affect the accuracy of the Runge-Kutta method?
2. What are the main advantages of using the Runge-Kutta method over other numerical methods like Euler's?
3. How does the error estimation work in the Runge-Kutta method?
4. Would you like to see the calculation of \( y \) at a different value of \( x \) using the same method?
5. Can we apply the Runge-Kutta method to systems of differential equations?

**Tip:** When using numerical methods like RK4, smaller step sizes generally result in higher accuracy but require more computational effort.

Apply Runge-Kutta fourth order method to find an approximate value of y when x = 0.1 given that dy/dx = x^2 - y, y(0) = 1.

Solve the differential equation dy/dx = x^2 - y using the Runge-Kutta 4th order method to find an approximate value of y at x = 0.1 with initial condition y(0) = 1. Learn the step-by-step solution with a detailed explanation of key concepts in numerical methods.

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