roblem $1[20\%]$: Find the Laplace transform of the following signals:

a$)()=2+4(),>0$

b$)()=2()+2$ sin$(2)2$ cos$(2),>0$

Recall: $()$ is a step function.

Problem $2[40\%]$: The switch has been open for a long time. At $=0$ the switch is closed.

a$)$ Determine the initial current $(0).$

b$)$ Find the differential equation for $()$

c$)$ Transform the equation into $$ domain and obtain $().$

d$)$ Determine $().$

Problem $3[40\%]$

a$)$ Use the linearity property to find the Laplace transform of $()=().$ Hint: recall

that $()=1$

$2(+).$

b$)$ Compute the Laplace transform of $(),$ when $()$

$+6()=4(),(0)=3.$

c$)$ Find the Laplace transform of the following signal $()=7(),$ where $()$ is a step

function