$($a$)$ Use Maclaumin series to solve the differential equation $y^{\text{'}\text{'}}=-y$ with initial conditions $y(0)=1$ and ${}^{\text{'}}(0)=0$

$($b$)$ If the initial conditions were instead $y(0)-0$ and $y^{\text{'}}(0)=1,$ when does that make the solution?

$($c$)$ Tet $y=p(x)$ denote your solution for $(a)$ and $y=(x)$ your solution for $($b$).$ Show that any function $yf(x)$ that solves the differential equation $y^{\text{'}\text{'}}-y$ can be written as $f(x)=ap(x)+bq(x)$ where a and $b$ are renl numbers. $($Hint: think shont splithing $f$ into its odd and even compotients.$)$