Question

QUESTION 10 d x dt2 -7x + y = 0 Use the elimination method to find a general solution to the given system. dy 7x + dt? -7y=e4t x(t) OA y(t) = x(t) ов. y(t) = 1 32 x(t) 32e4t+CZ+Czt-Cze 14t-ce-V14t 32@4t+C+c2t+Czevz4t-cze Ce-/141 32 @4t+CZ+Czt+Czev14t+Ce=V14t 24t+c+C2t+Czev 14t+Cge=V147 *+C+Cat-Czev14t -Ce-/147 €4t+c7+zt+Cze/14t+Cae 24t+C+Czt-Czev 14t-Cae (0) 14+C+CD+C;ev/ |X(t) =- 32°4*+c4+Cat-cz 24t+C+Czt-Cze/14t+cev ********(+6+(30/14-Ge/14 OC 7 - eft, 32 9 32 y(t) = 7 x(t) =- 32 14t - 14t 14t

Answer 1

Solna Given, equation, System of differential d's dt? (+7y = 0 day +7X – 7y 4t dt? - 7x + 7y 0 -2) el こ D h9g have. ct =0 (3) Replacing (627) x + Zy (62 -7)y We how eliminate u Multiply (4) by 7 - +7x eet - (4) as follows egn (3) byc (02-7) anel hee have

Date Page + 7 (D²-7) y 0 (5) D 2-7.) e a 7 (D² -7) y + 492 47 7 e - (6) -7 ett - 7 e46 or x = -zett or x - zett Subtracting (6) from (5), we have (D2 -7.2 x - 49x ox { 02-7)2 - 498x = {D4 + 48 - 14 D² - 49} x DP ( D2 - 14)} which a linear equation with Constant Coefficients Consider De (0²-14) x Auxilliary equation is given by D2 (D2 14) Soleing for a D D = 0,03 + Vit Required Root of Auxiliary equation. is O. = O. 7

Date Page Comp lementary room the function Corresponding to not of aurilli egulation is given by Cif - le + Cat) eott rieť Geriat + (4 ICF D or t at a t t ze eviat + Getinta

Particular of egn (7) is. Integral given by P.I= -7 0212-14 By using formula of formula of P.I. te TIE 1 eat eat = 1 f(0) f(a) f(a) o Where Therefore P-I = ett 7 (412 ( (4)2-14)

P-I = ett - 16 x 2 P.I ett 7 32 (9) General solution of egn (7) is given by x = Cof + P.I. ta C + Cat + z evi42 + C4 e-riot -7 ett 32 Ellos

Date Page we have Differentiating (10) w.xit .f? XU) = 0+ (q + Vit (z enige 1196.4 e-viAt- Differentiating (11) wort 't, we have 2" () 14 Cze + 14 cte 7 32 x 4 ett (11 rint VAT 7 2 ett 12 we have . + Nowe from egin) (1) Я = 1 {7. x"(t) Putting values of X and xilot) from (10) and (12) y = C, tent tez eritt -I eat + nee get toe-ritt 1 { 14 Cz eritt +1414e-rige .q ett G + Cat - Cz erist e-1147 32 7 2 CA + ett. ܠܐ 32 yt) = G, that - Ezeriut 4 e-viat + gett 32 -16

Date Page Hence, from (10) and (13) General Solution of System (1) and (2) is given by X(t) = C, that the pritet qe-1142 -7 ett 32 and ya) = Citlet - evitt-C. evitt tg e4t " Since are 32 ; and (4 arbitrary Constant we Replace y = -3 and (4= both equation. we get -7 + C, +Cqf - Cze 114 a 32 -ca in alt) ett - 24 e-114 € and h ett yt) + C + Cat +(3er14 t e -7144 32 +ca e option (C) is the option. Auss Idence, correct