Problem $1.$

Given: the nonlinear, time$-$invariant ODE

${}^{}-2{}^{}{?}^2+2sin=3tan{}^{}+2$

where $$ is the input signal in rad and $$ is the output signal in rad.

Determine:

$($a$)$ the equilibrium$/$trim input, $\frac{?}{\text{b}\text{a}\text{r}}(),$ for the equilibrium$/$trim output $\frac{?}{\text{b}\text{a}\text{r}}()=\frac{}{2}$rad;

$($b$)$ a transfer function model for the linearized, time$-$invariant system of the perturbed input and

output about the trim point in part $($a$)$; and

$($c$)$ the stability of the trim point based on the result in $($b$).$

Hint: you can assume the small angle approximation for $$ and ${}^{}.$