Question

DIFFERENTIAL EQUATIONS

Solutions of Systems of Linear Differential Equations

(L01.5 - 15 points) Show that the general solution of the nonhomogeneous linear system is x=(1 71}x+ []}<+[4.]e + 3' X=al_2-vzlevat +al-1 + vale-vēt + [1]{2+{2}]++ [4]

Answer 1

Given system 2X Com cestom *- J*- [1] **[*]:] - x'= Ax+fa), Say, then A=(-17 o fes= [i]++ [58]++ [ It's solution is of the from X=Xctxp. foo Xc find eigenvalues & eigeveckas Of A. for eigenvalues take det(A-DI) =0 Falco G1-() (1-7) – ED(-1)= + a²-1=0 02-2=0 0²=2 > D=152. ..a=-52, J2 are eigen values of a let a = +52, dat

foo eigenvectors take two d= 52 (A -J22)x=0 H-82 (A-AI) y=0, where x=[2.4) for --SZ (A-S27DY=0 [H2 -17 - 14 20 . Rg~ R2 + 14 ER (1+2 70 | - |- |x=0. R2 => 2, HR [ 0 ]--63 [f1=2) 2)-(87 lo 0107-C37 (0°)=(3) ( 72) 2-4-0, 0-3 --J51-9. CH2)2-970, oro | | GIt 2) 154. b= litus] is eigenectors of *B1- Limpel-above] | B37-cija] - [fusila - 01Five] is eigenwecker | A for da vz. Xc= 90, 291, ezt of A for a = vz | X=4 (iv) vividovat E ose method of undetermined coefficients Siece fara [!]++ [46]t+ [5], the free!

pasticolar solution is to= Xp=fait tattar P. Last²t2t+ cz) Since xp is solution of given System >> Xpl = [ 1740+ [1]2+[][+] - = Ei draittstta, y la, t²-bt-c, -cast bi7 I attatre, J = 1 read Last 66-ci tatt site, I + / +746-17 Lt²-66 +5l zaitt bi] Naraz)t +66, 52) t-9% tttt-17 azttbl LEG, taz)t²+ (be best at the tis! 3 ot²+za, ttb = Ga-G2+1)+²+ EQ-624426-6-6- چرا | 0.t? +20,476,- (actanti) + +63 +62-63t-Grets: quate coefficients of tstjlon both side - o--9-92+1, 29,=-2-buty, 6=-6-2-1 6 - 0=-9 taytl, 2a = 9+2 +6, b2 = -4+655. 10 to

Solving Co., arta_=1 9,-42=1 29, = 2 . =9,=1 Then toon 0, 03-1RaLA) oraz oraz ben ☺ ☺ becomes 2=-69bat4, 02-bitb2-6. . bitb2=2, . b-b2 =-6.. solving above two eas we get 6,=-2 Then 2=4. , © becomes -2--6-62-4, 42-479275. C+2=1, -iz-fthe solve, to get q=1, 2220 Thos xp = (si)t + [] [2] p= [:]++ [2]+[0] :. General solution of given system XC6) = *c+Xp. X= ( Livelet + C[itda] 2x24 [814+ [3J4+ [8]