Question

(3): Use any method of your choice to obtain a general solution for the differential equation given below. Assume zero initial condition. y + 4y + 4y = t + sin(t)

Answer 1

Griven ordinary differential equation is ÿ + 4y + 4y = t + sint day y do no le t + sint dta taga + 4y = lieu ( + 4D +4) y = t + sint ; Daddy Now Auxilary equation of 0 is given by 29+ 40+4 = 0 (0+2782 -0. 9 = -2,-2 So, Complementory function Cof: = (C1+Cat Deat and particular integral (P.I.) = (t + sint) - sint .

Now, I t = C D + 4D + 4 (0+2) C + (1+(9+9'))". + - -- (9+2°) + (9+gum ye.-Jot + / -1 + 0 + 7 (t-1) . . And . sint = Da + 40+4 . sint sint (3+ve Sint I ( By putting Do. - pp) E p + 40+4

i sint sint a - 40+3 sint / By substituting 0 -M - (Ota = (40-3) sint (48 + 3) (4D-3) (40-3) 1609-9 sint sint ( By putting i 40-3 16(-1)-9 - - (1999) (sine) s { 4 o(sint) – 3 sine] = ( 4 cost - 3 sint ? thus complete solution is y lat) = C.Fit PI. = (C1+Cot) &** [uast - 3sint]