Cauchy problem in Mathcad for the n-order differential equations with one unknown variable (ordinary differential equations) is defined the following way. Let’s find the solution to the differential equation:
In the form of function y=y(x), which obeys the assigned initial conditions
where - is the assigned value. Cauchy problem solution for ordinary second or higher-order differential equations can be restricted to a set of equations. Cauchy problem for ordinary first-order differential equations solution with various functions applied is shown in the programme listing.
The most widely used method for Cauchy Problem solution is Runge-Kutta method. The method is based on the successive finding of the desired value of function yi+1 as the following formula suggests
h is a sufficiently small step, with help of which the whole Cauchy problem interval is divided into discrete points, in which solution is being searches for. Result error is analogical to the fifth power of the step (h5).
The geometriv sense of Runge-Kutta method is as follows. Direction (angle) is chosen out of the immediate point (xi,yi) , for which tg()=f(xi,yi). Here the point is estimated with the coordinates of After that direction (angle) is chosen out of point (xi,yi), for which
Here the point with the coordinates is estimated. After that out of the point (xi,yi) the direction (angle) is chosen, for which
Here the point with the coordinates is estimated in Mathcad.
After that out of the point (xi,yi) the direction (angle) is chosen, for which All the four obtained directions are neutralized in compliance with the calculation . This resulting direction is the platform for the building of the design point with the coordinates
Runge-Kutta method, due to its high accuracy, is used when performing the figurative solution to differential equations, in Mathcad as well. There are a few varieties of this method which were illustrated in the studied above functions. In the programme listing you can both compare the results, gained with help of different functions and evaluate the results on accuracy of calculation.
The results of problem solution compared, it is possible to come to the conclusion concerning the accuracy of problem solution. The most accurate result can be obtained by using function Rkadapt.