$5.$ A spring mass system consists of a horizontal metal support rod, which has a spring attached to it so that the spring hangs vertically downward and hangs freely.

At the end of the spring a $2$ kg mass is attached. The spring constant for the spring is $\backslash (\backslash$mathrm$\{$k$\}=50\backslash$frac$\{\backslash$mathrm$\{$~N$\}\}\{\backslash$mathrm$\{$~m$\}\}\backslash ).$

External dampening forces, such as air resistance, friction, etc are cumulatively known to be $12$ times the instantaneous velocity of the mass and that this overall dampening force always acts in a direction opposite of motion.

Lastly, there in a periodic, outside force acting on the mass, given by $\backslash ($ F$($t$)=10\backslash$sin $(4$ t$)\backslash )$

a$)(2$pts$)$ Create the specific differential equation describing this mass' displacement with respect to time based on the given information

b$)(3$pts$)$ Calculate values for $\backslash (\backslash$lambda $\backslash )$ and $\backslash (\backslash$omega $\backslash )$ and determine whether this system will be under damped, critically damped or over damped before the external forcing is factored in$.$

c$)(1$pt$)$ If everything else remained the same, what value of $\backslash ($ k $\backslash )($spring constant$)$ would yield a situation where the system was critically damped?

d$)(6$pts$)$ Solve your differential equation from $($a$)$ if $\backslash ($ x$(0)=0\backslash )$ and $\backslash ($ x$^\{\backslash$prime$\}(0)=5\backslash )$

You may use any technique or formula within the book or that we've learned in class to tackle this problem