$($a$)$ A manufacturing process involves a force being applied to an mass$-$spring system.

The force follows the function:

The mass has a value of $5$ kg and the spring has a stiffness of $20$ N$/$m$.$

i$.$ Determine the complete general solution for displacement x$($t$)$ including the complementary

function.

ii$.$ Determine the exact solution for x$($t$)$ when the displacement at t $=0$ is $0\mathrm{.}5$ m$,$

and the initial velocity is zero.$($a$)$ A manufacturing process involves a force being applied to an mass$-$spring system.

The force follows the function:

$F(t)=-0\mathrm{.}5t^2+3t+2+5e^{-t},t0$

The mass has a value of $5kg$ and the spring has a stiffness of $20\frac{N}{m}.$

i$.$ Determine the complete general solution for displacement $x(t)$ including the complementary

function.

ii$.$ Determine the exact solution for $x(t)$ when the displacement at $t=0$ is $0\mathrm{.}5m,$

and the initial velocity is zero.