Question $5(20$ points $+10$ points $($Bonus$))$ A system is governed by the differential equation:

$x^{}(t)+x^2(t)=\sqrt[\phantom{2}]{u(t)}$

$($a$).(8$ points$)$ With $x(t)=x_o+x(t)$ and $u(t)=u_o+(t),$ where $x(t)$ and $(t)$ are small time

variations. Find the operating point $x_o$ for $u_o=16.$

$($b$).(6$ points$)$ Find the corresponding linearized model governing $x(t)$ and $(t)$ about the

operating point. $($Note that there are two nonlinear terms in the differential equation, the term

$x^2(t)$ and the term $\sqrt[\phantom{2}]{u(t)}.$ You have to linearize both in this case.$)$

$($c$).(6$ points$)$ Solve the linearized model in Part $($b$)$ with $(t)$ being a constant $C.$

$($d$).($Bonus$,10$ points$)$ Estimate the range of validity of your linearized model by looking at how

well the term $x^2(t)$ is approximated by a first order approximation at $x_{o.}.$ Then estimate the

range of validity by looking at how well the term $\sqrt[\phantom{2}]{u(t)}$ is approximated by a first order

approximation at $u_{o.}.$ Which one would you pick as range of validity for the linearized model?

$[$Hint: refer to the result of Part $($c$)].$