Free motion $-$ No Dampine:

$\frac{d^{2}y}{d{t}^2}9y=0,y(0)=2,y^{\text{'}}(0)=3$

FRee Motion $-$ Dampine

$\frac{d^{2}y}{d{t}^2}4\frac{dy}{dt}4y=0,y(0)=1,y(0)=2$

For motion $-1.$ Dampine

$\frac{dy}{d{t}^2}2\frac{dy}{dt}5y=2sin(t),y(0)=0,y^{\text{'}}(0)=0$