1. **Problem statement:** Use the 4th order Runge-Kutta method to solve the differential equation $$\frac{dy}{dx} = y - \frac{x}{y}$$ with initial condition $$y(0) = 0$$ at $$x = 0.2$$ using step size $$h = 0.2$$. 2. **Formula for Runge-Kutta 4th order:** Given $$y' = f(x,y)$$, the next value $$y_{n+1}$$ is computed as: $$ k_1 = h f(x_n, y_n) $$ $$ k_2 = h f\left(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}\right) $$ $$ k_3 = h f\left(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}\right) $$ $$ k_4 = h f(x_n + h, y_n + k_3) $$ $$ y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) $$ 3. **Apply to our problem:** - Initial values: $$x_0 = 0$$, $$y_0 = 0$$ - Step size: $$h = 0.2$$ - Function: $$f(x,y) = y - \frac{x}{y}$$ 4. **Calculate each slope:** - Calculate $$k_1$$: $$ k_1 = 0.2 \times f(0,0) = 0.2 \times \left(0 - \frac{0}{0}\right) $$ Here, division by zero occurs because $$y=0$$ in denominator. This means the function is undefined at initial condition. 5. **Conclusion:** The initial condition $$y(0) = 0$$ causes division by zero in the differential equation $$\frac{dy}{dx} = y - \frac{x}{y}$$ at $$x=0$$. Therefore, the Runge-Kutta method cannot be applied directly at this point. **To proceed, either:** - Choose a different initial condition where $$y \neq 0$$, or - Use a method to handle the singularity or reformulate the problem. Since the problem as stated is not solvable with the given initial condition, no numerical solution at $$x=0.2$$ can be computed using RK4 here.