Question

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system, $\dot{y}\left(t\right)+5y\left(t\right)=u\left(t\right)$
when $y\left(0\right)=1$ and u(t) is a unit step function, y(t) is

• A. $0.2+0.2e^{-5t}$
• B. $0.8+0.8e^{-5t}$
• C. $0.2+0.8e^{-5t}$
• D. $0.8+0.2e^{-5t}$

Answer 1

The correct option is C $0.2+0.8e^{-5t}$

$\dot{y}\left(t\right)+5y\left(t\right)=u\left(t\right)$

Taking the Laplace transform of the above equation
$sY\left(s\right)-y\left(0\right)+5Y\left(s\right)=\frac{1}{s}$

$(s+5)Y\left(S\right)-1=\frac{1}{s}$

$(s+5)Y\left(S\right)=\frac{s+1}{s}$

$Y\left(S\right)=\frac{s+1}{s(s+5)}$

$\frac{A}{S}+\frac{B}{(s+5)}=\frac{s+1}{s(s+5)}$

$A=0.2$

$B=\frac{-4}{-5}=0.8$

$Y\left(S\right)=\frac{0.2}{s}+\frac{0.8}{(s+5)}$

$y\left(t\right)=0.2+0.8e^{-5t}$