$238$

Transform Solutions of Linear Models

$*7\mathrm{.}11.$ Consider a first$-$order system with input $u(t)$ and output $y$ that is described by $y^{}+0\mathrm{.}5y=u(t).$

a$.$ Find and sketch the unit step response $y_U(t).$

b$.$ Find and sketch the response to $u(t)=2$ for $t>0$ and with $y(0)=-1.$

c$.$ Find and sketch the response to $u(t)=U(t)-U(t-2)$ with $y(0)=0.$

d$.$ Find and sketch the unit impulse response $h(t).$

$7\mathrm{.}12.$ The voltage source in Figure P$7\mathrm{.}12$ is zero for but has a constant value of A for all $t>0.$

a$.$ Find and sketch the output voltage $e_o^{}$ for all $t>0.$

b$.$ Check the steady$-$state response by replacing the capacitor with an open circuit.