Question

Use the method of undetermined coefficients to find a general solution to the system x'(t) = Ax(t) + f(t), where A and f(t) are given. - 117-1 -2-5] x(t) = 0

Answer 1

flt)=9t-5 where 16 -39t-7 Valuus A Fast we will fnd JA-AI -A -16 i-A) (-A) 16 |-A4 -5, A2-3 Eigen Values f A are Vectos osespenday to A5 us Egen (A -5エ) X X 5 6 1-5 4 16 2 4

Matnx Coefficient Alyng Re operalians on R2R 4RL 4 2 - Χ2 ч let 1 Cesespdeng to 5 is ugen Vectos (A31) - 3 Fo 6 4 RR-4R 2 TX - 2 41 let 24 to --3 Egen Vector leespen deng +e3t 1 Past is Sol to Homo geneous

let the patiaulas see" be of the form a bt Xpe whese a a. 9 b2 Axt flt), we get -at -5 -31t 7 ait bit b2 a2+ b2t a,t bit 16a, +16 bt a+ b2t +a2 t b2t -at- 5 3at-7 (a+a-5t (b, b- (169,+ a2-7 ) +t(16 b, t b2-33) b2 5+ t (btb2-2 ata2 ba 16a a2-7 + t(16b, +b2-32 Jet On Comparng ceriient of Lonstant 9 we b aita2-5 ba 2 16a,+ a2-7 b,t b 0 16b, t b-32- D On sebuung weget bt b2 6 b b2 3a 5b 30

ai+a2-5 -7 - 16a, a2 a,ta2 16a,+92=7 5a a D On seluing 2t Gensal sol to Axtf (t) i 3t 5t 9t t C2 e -1