12.16 LAB: Implicit Solution An equation is said to be "explicit" in some variable x if the equation can be rearranged to place x by itself on one side of the equals sign. Any equation that is not explicit in x is said to be "implicit in x". Often, we need to solve implicit equations. "By hand" implicit-solution approaches like Newton-Raphson are often taught to engineering undergraduates. Python also provides a convenient way to solve implicit equations using scipy.optimize.fsolve(). To see how scipy.optimize.fsolve() works, let's consider an example. Say you have the implicit equation: 1 1 = ce + 20.5 You may spend as long as you like trying to manually solve this equation for x...or you can do this. First, cast your equation as a function of X: 1 f(x) ce 1 + 0.5 In this form, the value of x you seek, which we can tag with an asterisk, satisfies f(x*) = 0 We can send this function to scipy.optimize.fsolve() and it will give us back x*: from scipy.optimize import Esolve from math import exp, sqrt def my_func(x): exp(-x*x) + 1.0/sqrt (x) return X 1 x_star = fsolve (my_func, x0=1) print('x_star {: .4f}; checking: f(x_star) = {:.45} .'. format (x_star[0],my_func (x_star[0]))) Here, the call to fsolve takes the function identifier as the first positional argument, and we specify an initial guess for x* using the keyword argument x0. fsolve returns a tuple, and the first element is x* The result: x_star = 1.4612; checking: f(x_star) = -0.0000. Write a Python program that will find the solution to =e* 22 tan(x) and print the result the same way as the example above. main.py 1