A circuit consists of a resistance $R,$ capacitance $C$ and constant electromotive

force $V.$ There is an initial charge $Q_0$ on the capacitor at $t=0.$ Let $q$ denote the

charge on the capacitor at time $t.$ Explain the origin of the differential equation

for $q$ shown here and solve it using a method of your choice:

$\frac{dq}{(d)t}+\frac{q}{RC}=\frac{V}{R}.$

Identify the behaviour of $q$ as $t$ becomes large in the solution and explain what

this means in terms of the physical system being modelled.

Let $f(t)$ be a function known for any time $t0,$ with a well$-$defined Laplace

Transform hat$(f)(s).$ Let $g(t)$ be a second function given by

$g(t)=e^{-at}f(t).$

Prove that

hat$(g)(s)=$hat$(f)(s+a).$

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Use this result to find the Transform of the signal $x(t)$ :

$x(t)=\{\begin{array}{ccc}m{e}^{-at}& \text{f}\text{o}\text{r}& tb\\ 0,& t>b.\end{array}$

Let $f(t)$ be a function known for any time $t0,$ with a well$-$defined Laplace

Transform hat$(f)(s).$ Let $g(t)$ be a second function given by

$g(t)=e^{-at}f(t).$

Then the Shift theorem states that

hat$(g)(s)=$hat$(f)(s+a).$

Use this result to find the Transform of the signal $x(t)$ :

$x(t)=B{e}^{-at}sin(t).$ please answer each part with calcuations thanks