Question

Find a general solution of the system x' (t) = Ax(t) for the given matrix A. 3 -- 1 A= 10 -3 x(t) = 0 (Use parentheses to clearly denote the argument of each function.)

Answer 1

given Systemy x'l4 = A xlt) aud A = 13 -3] So we can write x'lt 3 ) xit) 10 -3 So we can white Come on! (13 - 53 ) [] ye'ct) lyllti So 111 xilta 3 alt) – ylt) yllt) - Tout- 3 yCf) alow diffuentiale x"(H = 3x114) - yllf) Now substitute (2 yllt, 2"1419 3x'(+)-[10x1+1–3 y(+)] 2"CH) = 3x'14) - 10 21+ + 3 y(+) (3) from mit) - 32(+)= -41+) + 3 0467) y(+) -(4) So susstitute is (3) Odctcr agar al(t) = 3 x'+) -10 alt) + 3 (x1+3+376+1) x"(4) = 3x'(4) 107(4) - 321) +901f)

allt) 2 e to alltfalt aO (5) so this way and order homogeneary equation and Let cy assaue general solution (4= al and allesare %d si(4)= are the te So substitute ein (5) grestt est eft cont] =0 &0 So sit 120 I et T 22 gril 8,2= £i & a£iß of deo, 3= So when the roots are complex and imaginary then the general solution Gecerlße) & be sex ße) Geo cost the scent a Costta sunt at alfla a(4) 2 alt)

3 32141 - allt) C, cost + 2 sent a then alta - sunt tc cst C cost t 4 Now from (4) ylt So it r(t) = So y H) = 3 [c, cost to sist]-[-q sent + (Cost] 914) = 36, cost + 36 sint te, suit - Cecast ylt) = (36,-(2) Cost + (6,+3(2) sint 36-(2= 16 (+312 = 24 yu