CauchyProblem

 static void BuildCauchyProblem()
        {
            List<double> y = new List<double>();
            double x = 1.0;
            double h = 0.05;
            y.Add(CountYPrecise(x));

            Console.WriteLine("Euler's method: ");
            for(int i = 0; i < 10; i++)
            {
                Console.WriteLine("{0} \t {1} \t {2} \t {3}", x, y[i], CountYPrecise(x), Math.Abs(y[i] - CountYPrecise(x)));
                x += h;
                y.Add(y[i] + h *CountEulersFormula(x, y[i]));
            }

            x = 1.0;
            List<double> y2 = new List<double>();
            y2.Add(CountYPrecise(x));

            Console.WriteLine("\nImproved Euler's method: ");
            for (int i = 0; i < 10; i++)
            {
                Console.WriteLine("{0} \t {1} \t {2} \t {3}", x, y2[i], CountYPrecise(x), Math.Abs(y2[i] - CountYPrecise(x)));
                x += h;

                y2.Add(CountImproveEulersFormula(x, x - h, h, y2[i]));
            }

            x = 1.0;
            List<double> y3 = new List<double>();
            y3.Add(CountYPrecise(x));

            Console.WriteLine("\nPredictor-corrector method: ");
            for (int i = 0; i < 10; i++)
            {
                Console.WriteLine("{0} \t {1} \t {2} \t {3}", x,  y3[i], CountYPrecise(x), Math.Abs(y3[i] - CountYPrecise(x)));
                double Fx1 = CountEulersFormula(x, y[i]);
                double fx2 = y3[i] + h * CountEulersFormula(x, y3[i]);
                double tmp = CountEulersFormula(x + h, fx2);
                y3.Add(y3[i] + h * (Fx1 + tmp) / 2);

                x += h;

            }

        }

        static double CountYPrecise(double x)
        {
            return 7 * x * x * x + x * x + x;
        }

        static double CountEulersFormula(double x, double fx)
        {
            return 7 * x * x * x + 22 * x * x + 3 * x + 1 - fx;
        }

        static double CountImproveEulersFormula(double x1, double x2, double h, double y)
        {
            double newy = y + (h / 2) * CountEulersFormula(x1,y);
            double x = (x1 + x2) / 2;
            return y + h * CountEulersFormula(x, newy);
        }



        static void Main(string[] args)
        {
            BuildCauchyProblem();
        }