Question

1. A system is described by the following differential equation, Solve this differential equation for y(t) using classical technique, if (0)=0 and dv/dt(0) - 3. dv(t)/dt' + 3dv(tydt + 2v(t) - 0

please explain the classical method i haven't done this in a while

Answer 1

Civen d vlt) +3 dvLt) +2V (t) = 0 at at Civen Vol=0 dulo) -3 DE the above differential ervation is givenas. b vlt) + 3D VLt) +2V (t) = 0 (02 +30+2) Ult) = 0 here o = ď dt2 Drd at the roots of D+3D+h are D222,-1 (D+2) CD+) VL+2 20 We know that

if (ota) CD+6) Y() = 0 the general solution of ylt) is givenas YLt) = G. Cat Gebt fimularly -2t -t - 0 V(t) = Ge toe 6 if vlo) o substitute in ie at to V(o)= 0. -210) (0) vlo= Cie + Ce 0= tla differentiate ervation ) dule) ce?(1) + (-1) at Put to dulo) - Ge (-2)+cze 30C-2)467 6-1) at 3 = G (-2) + (-1) Cz 3 = -24 -

G = - in equation & Put - 3 = -2 (-(2) - Cz 3= 20 - Cz 3=62 Put en (273 in elvation & G=-3 2 3 4 (₂=3 Substitute G a Valurs ie inervation -at -t ult) = -3 +3 e. | w(t) = 3 (et - ezt)