$6$:$14$

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a$.$ Obtain the input$-$output relation $($differential equation$)$ describing the system. $x(t)=Au(t)=$ input voltage, $y(t)=Ri(t)=$ output voltage across resistance $R.(4$pts$)$

b$.$ Using this input$-$output relation $($differential equation$),$ draw the block diagram representation of the system between $x(t)$ and $y(t).(3$pts$)$

c$.$ Solve this input$-$output relation $($differential equation$)$ for $y(t)$ for $t0$ with initial current at inductor $i(0)=l_0$ and $x(t)=Au(t).$ Note that this $y(t)$ must be in the form of $y(t)=y_n(t)+y_p(t)=$transientresponse$+$steady state response$=$homogeneous solution$+$particular solution. $(6$pts$)$

d$.$ Then re$-$arrange or re$-$write this $y(t)$ found in part$-$c in the form of $y(t)=y_{z1}(t)+y_{2s}(t)=$zero$-$ input response$+$zero$-$state response, and specify $y_{iz}(t)$ and $y_{zs}(t)$ parts. $(3$pts$)$

e$.$ Explain$/$show that whether this system is

I. Linear or non$-$linear? $(2$pts$)$

ii$.$ Time$-$variant or time$-$invariant? $(2$pts$)$

iii. Stable or unstable? $(2$pts$)$

f$.$ Find the output $y(t)$ if the input voltage $x(t)=u(t)-u(t-10),$ and $R=1$Ohm,$L=2H,I_0=0$ Amp.$(3$pts$)$