Consider the forced harmonic oscillator described by the differential equation

$y^{\text{'}\text{'}}+y=g.$

For let $g(t)=0,$ so that $y(0)=0$ and $y^{\text{'}}(0)=0.$ At $t=0,$ the forcing is instantaneously increased to $g(t)=\frac{1}{2}$ and maintained at this level until, at the later time $t=c,$ the forcing is instantaneously increased to its final value $g(t)=1.$

$($a$)$ Use the Laplace transform to find the solution $y(t)$ assuming $c>0.$

$($b$)$ How should you choose $c$ so that the system is in a steady state $($i$.$e$.,$ does not oscillate$)$ for all $t>c?$