Question

Consider the initial value problem y" +3y' +2y = (t-1)+r(t), y(0) = y(0) = 0, where 8(t-1) is Dirac's delta function and S4 if 0<t<1 r(t) 8 if t > 1 (a) Represent r(t) using unit step functions. (b) Find the Laplace Transform of 8(t-1)+r(t). (c) Solve the above initial value problem. {

Answer 1

Page Nord r(t) = 4+ if ost<12 if t> 8 ret) H 4 t r(t) = 4+ ult) – 4 (4 -2) u (1-1) +44 (+-2) HitHH 1 4 4 u(t)-4 (1-2) u (t-1) 4+uct) X(t) 8 HA 4 V. 2 t YH) = ut ult) --4 (-2) (4-1) +44(+-1)

Page No.2 LT of St-1)+r(t) 5(+-1) + 4+u(t) - 4 (4-2) u(t-1)+4 (t-1) Laplace Transform LT = € S + 4 4 es $2 + 4 S 32 @y+3y + 2y = 84-1)+r y+3y'+ 2y - 56-1)++ (d) -4 (-1) -1) 1446-1) 49(0) = 9'6)=0 laking Laplace Transform of both side, [s2 y (5) -sy(0) - YO)] + 3 [sys) – syo)] +298) est 4 -4es 52 +46-5 S YO) = 0,7")=0 524 +35+2) y(s) = est 4 Чеѕ thes 82 s2 S Y(s) 4 4e's (S+T) (5+2) s? (5+1) 6+2) s² (5+1) (5+2) Чеѕ S(S+1) (st2

Page Nory y(s) - -s es + 5 2 8 1 St ? + 4 St1 St2 52 st2 [2es -5es + 4ers les S 52 sti st2 + 2 e-s thens St1 S42 Ihverse L. Transfer 964) > [e-4-1) 4(4-1) ----26-1) 21 (5-1)] [2u(t)- 5tu(t) + Hetu(t) - x-2t ult) 2t ult) - [ 2u (t-1) - 5 (t-1) e (+-2) +4e-Gt-1)(4-1) - 4-2 (4-1) 4(t-1)] + [12 (4-1)-4-4-1)+86-4-2)] u (t-1)