Question

Let a causal LTI system be characterized by the following differential equation, with initial rest condition.
$\frac{d^{2}y}{d{t}^2}+7\frac{dy}{dt}+10y\left(t\right)=4x\left(t\right)+5\frac{dx\left(t\right)}{dt}$

Where, x(t) and y(t) are the input and output respectively. The impulse response of the system is [u(t) is the unit step function]

• A. $2e^{-2t}u\left(t\right)-7e^{-5t}u\left(t\right)$
• B. $-2e^{-2t}u\left(t\right)+7e^{-5t}u\left(t\right)$
• C. $7e^{-2t}u\left(t\right)-2e^{-5t}u\left(t\right)$
• D. $-7e^{-2t}u\left(t\right)+2e^{-5t}u\left(t\right)$

Answer 1

The correct option is B $-2e^{-2t}u\left(t\right)+7e^{-5t}u\left(t\right)$

$\frac{d^{2}y}{d{t}^2}+7\frac{dy}{dt}+10y=4x+5\frac{dx}{dt}$

$(s^2+7s+10)Y\left(s\right)=(4+5s)X\left(s\right)$

$\frac{Y\left(s\right)}{X\left(s\right)}=\frac{5s+4}{s^2+7s+10}$

Impulse response = $L^{-1}$(Transfer function)

$=L^{-1}\left[\frac{5s+4}{(s+2)(s+5)}\right]$

$=L^{-1}[-\frac{2}{s+2}+\frac{7}{s+5}]$

$=-2e^{-2t}u\left(t\right)+7e^{5t}u\left(t\right)$