Calculate by the Runge-Kutta’s method integral of the differential equation at the initial condition on the segment [0, 0.5] with the step of integration S o l u t i o n. Let’s calculate . For this purpose at the first we’ll consistently calculate :
Now we receive:
and, consequently,
The subsequent approaches are similarly calculated. Results of calculations are tabulated:
Results of numerical integration of the differential
equation (1) by the fourth order Runge-Kutta’s method
So
,
For comparison the exact decision of the differential equation (1) is:
whence Thus, exact and numerical solutions of the equation (1) have coincided up to the fourth decimal place. The fourth order Runge-Kutta’s method also is widely applied to the numerical solution of systems of ordinary differential equations. |