Question

A control system is represented by the given below differential equation,
$\frac{d^{2}y}{d{t}^2}+5\frac{dy}{dt}+4y=x\left(t\right)$
has all initial conditions as zero. If an input x(t) = 8 u(t), then output y(t) is,

• A. $[1-\frac{1}{4}{e}^{-4t}+\frac{1}{2}{e}^{-t}]u\left(t\right)$
• B. $[1+\frac{1}{4}{e}^{-4t}+{\frac{1}{3}}^{e-t}]u\left(t\right)$
• C. $[1+\frac{1}{8}{e}^{-4t}-\frac{1}{3}{e}^{-t}]u\left(t\right)$
• D. $2[1+\frac{1}{3}{e}^{-4t}-\frac{4}{3}{e}^{-t}]u\left(t\right)$

Answer 1

The correct option is D $2[1+\frac{1}{3}{e}^{-4t}-\frac{4}{3}{e}^{-t}]u\left(t\right)$

$\frac{d^{2}y}{d{t}^2}+5\frac{dy}{dt}+4y=x\left(t\right)$

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

$x\left(t\right)=8u\left(t\right)orX\left(s\right)=\frac{8}{s}$
$(s^2+5s+4)Y\left(s\right)=\frac{8}{s}$

$Y\left(s\right)=\frac{8}{s(s^2+5s+4)}=\frac{A}{s}+\frac{B}{s+4}+\frac{C}{s+1}$

$Y\left(s\right)=\frac{2}{s}+\frac{2}{3(s+4)}-\frac{8}{3(s+1)}$

$y\left(s\right)=2[1+\frac{1}{3}{e}^{-4t}-\frac{4}{3}{e}^{-t}]u\left(t\right)$