Problem $4(20$ points$)$ Consider the equation $y^{\text{'}\text{'}}-y^{\text{'}}=0.$

$($a$)(10$ points$)$ Show that $y=C_1{e}^x+C_2$ is a solution to the equation for any constants $C_1,C_2.$

$($b$)$ points$)$ Find $C_1$ and $C_2$ that satisfies $y(0)=10$ and $y^{\text{'}}(0)=100.$