(Solved): In Python The driven and damped oscillator has the solution (for the under-damped case): x(t)=xd ...

In Python

The driven and damped oscillator has the solution (for the under-damped case): $$x(t)=x_d(t)+x_f(t)$$ $$x_d(t)$$ is the damped solution calculated above, and the additional term $$x_f(t)=A_f\cos \left({?}_{f}t+{?}_f\right)$$ is due to a sinusoidal driving force of amplitude $$f_0$$ and angular frequency $${?}_f$$, i.e. $$f(t)=f_0\cos \left({?}_{f}t\right)$$. The parameters $$A_f$$ and $${?}_f$$ are given by $$\begin{array}{rl}{A}_f& =\frac{f_0}{\sqrt{{\left({?}_0^2?{?}_f^2\right)}^2+{?}^2{?}_f^2}}\\ {?}_f& ={\tan }^{?1}\left(\frac{??{?}_f}{{?}_0^2?{?}_f^2}\right)\end{array}$$ Create two functions with suitable parameters - one to calculate $$A_f$$ given $$f_0,{?}_f,{?}_0$$, and $$?$$ - and another to calculate $${?}_f$$ given $${?}_f,{?}_0$$, and $$?$$. You will need to use the function np.arctan2 for $${\tan }^{?1}(x)$$. Hint: check the order of arguments of np.arctan2 in the documentation very carefully. - Test your two functions: work out some values on paper that should be returned and make sure your function returns them.