Question

A LTIC system is specified by the equation

(D^2 + 5D + 6)y(t) = (D^2 + 7D + 11)x(t)

Find the zero-input response of the response y(t) and the impulse response h(t) if the initial conditions are y(0) = 0 and y'(0) = 1.

Answer 1

sol: Guiven LTIC system is given by (0+50+6) y(t) = (D+90+1)x(+ Initial conditions are yo) =0) { y'o =) Zero Il Response (5+50 + 6) yet) = (5 + 7D + 1D Alt Characteristic equation is +5+6=0 X + 2x+32 +6=0 (a ta (2+3)=0 d=-2, -3 As roots are unique, Characteristic Modes are e2t, e-3t so yo(t) = ce at +6 e 3t Given Conditions are yo)= 0, '10) = 1 Yo (o) = C, e210) + C2 e 310) o=G+C2 o Yoct) =-20, e 2t - 3egest using y'o)=1 1 = -20, -302 - © Multiply eq 0 with eq @ & solving ay

20, +26₂ = 0 -22,-36,-1 - C₂=1 [C2 =-11 1. Yo (t) = (ezt - p3t) / ® Impulse Response: we have hlt) = bn 8(t) at [P (D) yolt) ult) 910) PD t bn=bo] Griven co +50 + 6)y (t) = (5 + 7D + DACE Ď term - bn ie (n=2) It is a and order Here, Coefficient of differential equation i, n=2. Now look at right side en bnai Now D+50 +6 s chooracteristic equation By solving we get roots -2 -3 as earlier Yo(t) = Ke-2t + k2e-3t. Here we take Ki & ka as eve already took C, & C2 in zero desponse. Yo '(t) = -2k1 e 2t_sk est Now using Initial conditions y col=0 y'(o): 1

Y you = kitk2=0 y'(o) = -2k, - 3K2=1 Now wing initial Now sowing above two, we get K1=1 & K2=-1 Yn (t) = ezt _ eat P (1) Yo(t) = (5 +70+ 1) (cr2t_x - 327 2 Cent, est ) + 7 d Cert est) + et est = a 2 (- 20 - 2t + 30 3t 9 + 7 (-2 & 2t +30-at, te-at e at. = 4e-2t - 9 - 3+ + 280 et 63 e 3t & pat _est P (0) Yalt) = 33 0 2t - 7303 So, Tmpus ®ал ролле. h(t) = 18ct) + [33 e 2t _ 73 4 3t Jult) het) = S(t) + [33 e 2t - 13 0 - 3t Juct) of
here bo=bn . If impulse response of a system exist for t>0 then system is said to be causal.