To obtain other values of
, we may simply solve
For example (see Fig. 1), solving
for gives It is in agreement with the fact that
In Fig. 2, we compute
from repeatedly solving for where
is a periodic function, has infinitely many solutions. It is possible that the solution obtained by Newton’s method lies outside of , the range of by its definition. Such solution cannot be considered the value of
Exercise-1 Compute by solving for
Exercise-2 Explain Fig. 3.
We have defined function
as a set:
It means that
is the unique solution of
, we solve for as follows:
from to gives
, 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
Exercise-1 Compute for .
Exercise-2 Explain why ‘rk’ cannot solve .
Instrumental Flying“, we defined as the inverse of and repectively.
To find the derivative of
(see Exercise-1) and (see Exrecise-2),
Similarly, to find
, since (see Exercise-3).
Exercise-1 Show that .
Exercise-2 Show that .
Exercise-3 Show that
Exercise-4 Differentiate directly (hint: see” Deriving Two Inverse Functions“).
An oasis awaits
The above image is created by
Omega CAS Explorer:
The governing equations are: