please show all work and steps

a$.)$ Given the solution, show that the solution satisfies the problem $($as seen in image$)$ tion

$x(t)=\frac{F_0}{{}^2-{}^2}(cost-cost)$

Show that the solution satisfies the problem,

$\frac{d^{2}x}{d{t}^2}+{}^{2}x=F_{0}cost$

b$.)$ Show that the solution satisfies the initial conditions, $x(0)=0,x^{\text{'}}(0)=0$