Runge-Kutta Method

Write a program for the second order Runge-Kutta Method with automatic step control to solve ordinary differential equation systems.  Store the results so that they can be used later to construct a table and to draw the curve as y(t) versus t or in systems y_1 versus y_2. Test the program with the problem y_1^’=-y_(2,) ? y?_2^’=y_1 with initial conditions ? y?_1 (0)=1,? y?_2 (0)=0. Verify that the exact solution represents uniform motion along the unit circle in the plane ?(y?_1,y_2). Stop the calculations after 10 revolutions (t=20?). Perform experiments with different tolerances and determine how small the tolerance should be so that the circle on the screen does not become thick. When applying the program to solve a second order equation, in the resulting system the unknowns represent the solution and of the original equation and its derivative y^’. Consider 6 values of the solution and construct an interpolation polynomial and show that the roots of y^’ are points of possible y-end.  TP:  Implement all the root search algorithms studied in classes.