Question

Chapter 7, Section 7.5, Question 20 Solve the given system of equations. Assume t 0 ty Hint: The system tx - Axis analogous to the second order fuer equation. Assuming that X-&', where is a constant vector, and I must satisfy (A-DE- On order to obtain rontva solutions of the given differential equation

Answer 1

The given system of equation is, tx'= 5 ( ) 7 t>0. 24 - 5 ti) cue have ro cue Knoco solve the given system of equations. that, the system tx'= Ax-ü) analogous second orden Eulen equation (ü) is to the cue can assume that x = & tro where & is a constant vecton and ¢ and a must satisfy (A-n I)&= 0 - in order to obtain non-rivial solutions of the given system Now inorden to obtain a non - trivial soulton & to given A-nIl = 0 [From ci] Fon the given problem, Goog The given problem (i) is of the form of (u). so, from () cul have 5-n -1 24 - 5-7 (5-1)(-5-n) +24 - 0 ~, -(5-0)(5+) +24=0. -( 25 - 12) +24 = 0 OT [:(5+r) (5-m)= 25-7]

- 25 +82 +24 = 0 m2-1 = 0 m²=1 a m+1 Thus, we have n=1 and m2 = -1 Noco , que have to determine the corresponding s. corresponding Let &s, be the constant vecton tom, :1 The from Civ we have (A-1.I) &, = 0 (0) (0) (w $•()] :)(:)-(0) a lot to) - ) 4 24 -6 401-B, =0 and 290,-63,=0 ov, 40,- Bi=0 1401, = BI since d, and ßi are arbitrary satisf atisfying the relation 4015 Bil Let Chouse di=1 then have ß = 4 Hence, & c)

Again, rer 92 - Contato the constant veeton corresponding to T = -1 Then from we have. (A-(-1)I) =0 GY 5+1 24 )(*:)-(3) - 5+1 02 ß2 24 -4 O 64₂-82 24d2-432 62-B2=0 and 2442-432 = 0 OY 16 d₂ = 32 o, 602 - B2=0 Since , and be are artimary satisfying kd=82 Thus we can choose de-1 such that F2 = 6 sol 3,- (6) The = G &,t"+ where ) 41 and cine general solution of the system (i) is given by +&+"2 arbitrary constants or, r=40) t'te (6)+1 4 (1) ++62(6+ arbihany constants. 2-4 where a and are