Question

Solve the following differential equations and find (a) Free response; (b) Forced response; (c) Steady state response; (d) Transient response; and (e) Total Response: 1. 4x + 7x = 6te -5t 2. 5x + 20x + 20x = 28, 3. 3 + 14x + 58x = 1740, x(0) = 5 x(0) = 5, (0) = 8 x(0) = x(0) = 0

need number 2 and 3.

Answer 1

2sx +203 + 20% = 28, 210)5, m?+ 4m+420 The arexiliary equation, sm+20m+20 20 (M+2) = 0 m= -2,-2 The complementary solu X(E)= e, e 2t + late at particular Solcetion (Steady State) solution is implementary 98204 . 28 cola хр(E)- - 5102+40+4) 5102 +410)+4) ХҮН Hence , the solution , XCEJ= p (t) + X(t) ni

2(E)= e, e 2t +cate2t + selt) = -2.c, et te ezt ac, te 20 X(055 - C, +(2(0) + 7 = 5 (0) 8 C 2 8o est. lot 8=33+ 8 xa- Creneral solution is xe (t) = 1 e 2 + thote t + 7

97 free response : The equation becomes. Cx+20 X +20 4:0 The solution is X(t) = xp(E)= e, e 2t teztezt 2105 C, C1) +6210 = 5 - 0,25 81E)= selt, et ezt xict) = -10026 + C₂ e 2t2l2t ezt

10)=8 8 = -10 (1) +2(1) - 0 -260 c2=18 X (E)= sê 20 tiste XiEJ= 50 2t +18te 2t 67 forced response, să + 204 +201228 X(t)=422 24+ mesté zlo C) steady response XIE)- Negleel the exponential term d) Transient response. 214)=18.62€ + Peg ze ZE e) Total response x6 y to