$2.$ It is a lesser$-$known fact that a Roomba $($robotic vaccuum$)$ cleans a house at a rate proportional to the amount of the house $($in square feet$)$ that is dirty. Assume that a $2100$ square foot house starts entirely dirty when you turn on the Roomba. You check back later and notice that the house is three$-$fourths clean after $\backslash (3\backslash$ln $(4)\backslash )$ hours.

$($a$)$ Set up a differential equation to represent $\backslash ($ x$($t$)\backslash )$ the square footage of the house that is clean at any point in time. You must solve for all of the coefficients in this equation. $($You may need to solve the equation to do that.$)$

$($b$)$ Now, we add the fun part of this dynamic. Assume that we add a dog to the house that dirties the house at a rate of $50$ square feet per hour. Set up a new initial value problem to model this situation with the dog involved.

$($c$)$ In the scenario in $($b$),$ what is the cleanest the house gets?

$($d$)$ Find an expression for the amount of time it will take for $\backslash (80\backslash \%\backslash )$ of the house to be clean for the situation in $($b$).$