We have defined function as a set:
It means that is the unique solution of
To compute , we solve for as follows:
Integrate from to gives
To obtain , we numerically evaluate , using function ‘quad_qags’.
The result is visually validated in Fig. 2.
Note: ‘romberg’, another function that computes the numerical integration by Romberg’s method will not work since it evaluates at
An alternate approach is to solve as an initial-value problem of ODE using ‘rk’ , the function that implements the classic Runge-Kutta algorithm.
Fig. 4 for
Putting the results together, we have
However, we cannot solve using ‘rk’:
Exercise-1 Compute for .
Exercise-2 Explain why ‘rk’ cannot solve .