Question

Consider a causal LTI system characterised by differential equation $\frac{dy\left(t\right)}{dt}+\frac{1}{6}y\left(t\right)=3x\left(t\right)$. The response of the system to the input $x\left(t\right)=3e^{(-t/3)}u\left(t\right),$ where u(t) denotes the unit step function, is

• A. $9e^{-t/3}u\left(t\right)-6e^{-t/6}u\left(t\right)$
• B. $9e^{-t/3}u\left(t\right)$
• C. $54e^{-t/6}u\left(t\right)-54e^{-t/3}u\left(t\right)$
• D. $9e^{-t/6}u\left(t\right)$

Answer 1

The correct option is C $54e^{-t/6}u\left(t\right)-54e^{-t/3}u\left(t\right)$

The differential equation,
$\frac{dy\left(t\right)}{dt}+\frac{1}{6}y\left(t\right)=3x\left(t\right)$

$So,sY\left(s\right)+\frac{1}{6}Y\left(s\right)=3X\left(s\right)$

$Y\left(s\right)=\frac{3X\left(s\right)}{(s+\frac{1}{6})}$

$X\left(s\right)=\frac{3}{(s+\frac{1}{3})}$
So,
$Y\left(s\right)=\frac{9}{(s+\frac{1}{3})(s+\frac{1}{6})}=\frac{54}{(s+\frac{1}{6})}-\frac{54}{(s+\frac{1}{3})}$

$So,y\left(t\right)=(54e^{-t/6}-54e^{-t/3})u\left(t\right)$