Question $3$: Consider the maglev system shown below, with the nonlinear dynamics given by $m{y}^{}+k\frac{u^2}{y^2}-$

$mg=0.$

a$)$ Write the non$-$linear state equations in the form ${\stackrel{}{x}}^{}\frac{?}{b}=ar(f)(\stackrel{}{x}\frac{,}{b}ar(u)),$ where $x_1=y$ and $x_2=y^{}.$

b$)$ Solve for all the equilibrium points.

c$)$ Find ${\stackrel{}{x}}^{**},u^{**}$ for the operating $($equilibrium$)$ point where $y^{**}=2.$

d$)$ Find the linearized state equations ${\stackrel{}{x}}^{}=A\stackrel{}{x}+B\frac{}{b}ar(u)$ at the operating point from part c $.$