Question

A solution for the differential equation $\dot{x}\left(t\right)+2x\left(t\right)=\delta \left(t\right)$ with initial condition x(0) = 0 is

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

Answer 1

The correct option is A $e^{-2t}u\left(t\right)$

$x\left(t\right)+2x\left(t\right)=\delta \left(t\right)$
Taking laplace transformation on both side
$sX\left(s\right)-X\left(0\right)+2x\left(s\right)=1$
$X\left(s\right)(s+2)=1$
$X\left(s\right)=\frac{1}{s+2}$
$X\left(t\right)=e^{-2t}u\left(t\right)$