Question

QUESTION 11 Find the solution of x' + 2x' +x=f(t), x(0)=1, x'(o=0, where f(t) = 1 if t< 2; and f(t) = 0 if t> 2.

Answer 1

A solution given x" +Qx! + x = Elt), X10 )= 1.X110: 0 Whenne fit)=(1 if t22 ♡ o if t 22 0 01 => Ft) 2 dx + X + dt² dt equation cel'i'th Cans dont this qe Concach differential Coefficient, the genellol Solletion = cofo & Polo of equho Is Thees on atibitary 02 Wheden co fol com plementary function involves ani bitary constant and poyo / Positicules integral) does not involves ane Constant Now equn con be written as (024 2D + 1) = Fits Wheme O'= the auxilany egun of 2 (024 20+1) 024 20 + 1 = 0 02 + 0 + 0 + Eu 0 0 + 1) + 2 CD + d ) = 0 =) (0+1) 1041):0 es and finding Cofo

D+1=0 & D+ 120 - D = -1:-1 MJs-dif Mas -1 300ts of quxilhoy equn @ Os some Corresponding part of cofo Pe of the form as Cie mitt tq e Hence equn Coro is of the form is Coro Crét + t Get tq e mat CP. J.) we finding particular int egral 1 (+) Poto : (02+20+2) fit)= 1 if tx2 then When 1 1 PI (024 20+ 1) using lale ot 1 eot 11 50 P.I. 02, 2014 using these property 1 ear FID) ax of cato e Fla)

So P. I. 1 eot 024qxott P.IE 1 . eot 0 +0+1 P.IS A {eote 14 When Fic)= 0. if tzo then P. I oo 1 D2420+1 P. I = 0 B Hence substuted the value of equn Ⓡ. eguna rand equh ® in equn o egun 0 and we get solution of equh @ X = cietyt get + 1-③, if, f(t) = 1, 242 and X= Ciet totet +0-0, f/10)20. tr2 Boundary using condition X10)= 1. fx110zo X(0) = 1. From equh @ we get Ce-o + Ox (2 e Ott 1- & t= Cit Cq xo tt Citd * La (CEO

From eguh ③ we get X; t Oxe + t Gei{ +1 Cgt e-t at X il n diff. equh. 0 wor. to. t, we get X' Co [ ette-t] to Cg [ e-tatet using X!(0)=0 Cq [ ero o e-o Cg [ 1-0) 1920 O- 02 we get From e quh ☺ Xz O, te- + [X=1 03 if flt) = t. +22 ergoin using Boundary condition X1072 1. 4x!(0)zo in equh 0 get we X(012 t 1= Ge-ot +G0e-o Iz Cito & rei= 1

From equh we get X 1. e-t t X e-t + Gg te-te a te-t - © . 7 diff equn © w. r. to 't we gel x's Let + ( [e't-tet? →x'=net + C2 [e-t-te-t] using XI10): 0, we gel e-o + [e-o-oe- - 1 + [-] 02

O= -1 teq Cq z 1 X = e-t t dit ect 07, if flt=0.622 X e-te t.te-t Hence complete Solution using Boundlahy Condition, we eget., From equo fequ" +22 X(t)= é-tt tect t 72 1 {extiel this is required solution Than ko