using System;

​

class Program

{

    static void Main()

    {

        // Set the initial conditions and step size

        double x0 = 0.0;

        double y0 = 1.0;

        double y1 = 0.0;

        double y2 = 0.0;

        double h = 0.1;

        double xMax = 1.0;

        // Solving the third-order linear homogeneous ordinary differential equation:

        // y'''(x) + 2y''(x) - y'(x) - 2y(x) = 0

        // Using the Runge-Kutta method for numerical approximation

        Console.WriteLine("x\t\ty");

        for (double x = x0; x <= xMax; x += h)

        {

            Console.WriteLine($"{x}\t\t{y0}");

            double k1 = h * F(x, y0, y1, y2);

            double k2 = h * F(x + h / 2, y0 + k1 / 2, y1, y2);

            double k3 = h * F(x + h / 2, y0 + k2 / 2, y1, y2);

            double k4 = h * F(x + h, y0 + k3, y1, y2);

            y0 += (k1 + 2 * k2 + 2 * k3 + k4) / 6;

            // Update y1 and y2 based on the derivatives

            y2 = y1;

            y1 = k1;

        }

    }

​

    static double F(double x, double y, double yp, double ypp)

    {

        // Define the third-order ODE: y''' + 2y'' - y' - 2y = 0

        return -2 * ypp + yp - 2 * y;

    }

}