Use the Laplace transform method $to$ solve the differential equation subject $to$ the

initial conditions $x(0)=1,x^{\text{'}}(0)=0,x^{\text{'}\text{'}}(0)=0\mathrm{.}5$ :

$x^{}+2x^{}+25x^{}+50x=0$

$x(t)=\frac{1}{58}[e^{-2t}cos(5t)+\frac{2}{5}{e}^{-2t}sin(5t)]$

$x(t)=\frac{1}{58}[e^{-5t}-\frac{2}{5}{e}^{-2t}sin(5t)]$

$x(t)=\frac{1}{58}[t{e}^{-2t}-e^{-2t}cos(5t)+\frac{2}{5}{e}^{-2t}sin(5t)]$

$x(t)=\frac{1}{58}[e^{-2t}-cos(5t)+\frac{2}{5}sin(5t)]$