Question

7. Given the non-homogeneous linear system of differential equations ? ′ = −2? − 7? + 3?

?′=−? +4? +?^-6t

1. Find its homogeneous solution using the eigenvalue-eigenvector approach (10pts)

2. Use the variation-of-parameters method to find its particular solution (10pts)

Answer 1

n = 3t C 22 241 2 - 7 4 eigen vectue of A are AdIl co 2 -2-d - 7 url. - 1 u-d solving d=-3, da=5 eigen vectors. (A-d, I W=0 [ Jaco 0-17) Simik anks (A-da I) 0 =0 v2=0 02-13)

solution of Homogeneous part is net- ciest * ()+coest () * 2 How let e 37 -est é 3t 5t e est e5t e 3t 8 e at 76 3t 3t 3t 8 8 - è 5t ہے ؟ 8 3t Xb= 8 3t e 8 3t -e 5t 8 6t -5t že 5t 1-3 te 7 et 8 theory Josh. * Cena - test zent dt lest I et -3t که م 24 I test Чо + 3 e 5t 200 I ellt 88 Particular solution = x je belt

UO solving xp(t)= 75 1472 448 66 e 4800 2113 128 4800 256 2112 e-6 79