Consider the following first$-$order differential equation:

$\frac{dy}{dt}=(t^2+1)y.$

$($a$)$ Determine if the given differential equation is separable. If separable. rewrite it in the form $\frac{dy}{d!}=g(t)h(y).$ If not, explain why.

$($b$)$ Find the general solution $y(t)$ to the differential equation.

$($c$)$ Verify if any particular solution $y(t)=k{e}^{t^{\frac{3}{3}}}$ satisfies the given differential equation. If it does, determine the value$($s$)$ of $k$ that work.