Example $2$: A CSTR $({}_1=10(s))$ followed by another $CSTR({}_2=2(s)).$

${}_1\frac{d{C}_1}{dt}+C_1=C_{in}$

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

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

occuring at $t=0$ s$.$

$C_{in}(t)=\{\begin{array}{cc}0\text{;},& t<mn\; id="EditorElement\_1341902">0</mn><mi\; id="EditorElement\_1341904">s</mi>\\ 1\text{;},& t>0s\end{array}$

Solve by using characteristic polynomial and Laplace transform. The answer is given as C$2($t$)=1-(1/8)*[10*$exp$(-$t$/10$s$)-2*$exp$(-$t$/2$s$)]$