Question

Two right sided signals are related through differential equations $\frac{dx\left(t\right)}{dt}=-2y\left(t\right)+\delta \left(t\right)$ and $\frac{dy\left(t\right)}{dt}=2x\left(t\right)$then the initial value signal x(t) is ____

• A. 1

Answer 1

The correct option is A 1
Taking Laplace transform of both sides of the two differential equations

$sX\left(s\right)=-2Y\left(s\right)+1$

and $sY\left(s\right)=2X\left(s\right)$

$sX\left(s\right)=-2\frac{2X\left(s\right)}{s}+1$

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

from intial value theorem,

$x\left(0\right)={\underset{s\to \infty }{lim}}^{}sX\left(s\right)$

$={\underset{s\to \infty }{lim}}^{}\frac{s^2}{s^2+4}=1$