Given the differential equation

$10y^{\text{'}\text{'}}+4y^{\text{'}}+\frac{y}{2}=0$

$(a)$ Show that $ify=e^{kx}is$ a solution $to10y^{\text{'}\text{'}}+4y^{\text{'}}+\frac{y}{2}=0$ then $10k^2+4k+\frac{1}{2}=0.$

$(b)$ Let $k_1$ and $k_{2}be$ the complex roots $of10k^2+4k+\frac{1}{2}=0.$

Find the solution

$y=c_1{e}^{k_{1}x}+c_2{e}^{k_{2}x}$

$to$ the differential equation

$10y^{\text{'}\text{'}}+4y^{\text{'}}+\frac{y}{2}=0$