Example $1$: A PFR $(t=10s)$ followed by a CSTR $(=5s)$

$C_1=C_{in}(t-t)$

$\frac{d{C}_2}{dt}+C_2=C_1$

To compute the step response, assume that $C_{in}(t)$ is a unit

step occuring at $t=0s.$

a$*$exp$[-$t$/5$s$]$ comes from general solution. Explain how these C$1($t$)$ and C$2($t$)$ are obtained by characteristic polynomial.