Question

Consider the following system. = x + y - 2 ot dy at = 5y = -2 at Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) 2= Find an eigenvector for the corresponding eigenvalues. (Enter your answers from smallest eigenvalue to largest eigenvalue.) K = K = K = Find the general solution of the given system. (x(t), y(t), z(t)) =

Answer 1

Sol. 2+y-Z 장성 2 - → Elf Sy OS O y - dy dt dz at z' - Y - Z Z O So Here A: s LO la) for eigen value , TA-VI) = 0 1-d s-d -0 (1-1)(S-1)(-1-1)-0 I-d -1 (0-0)-1(0-0)=0 O O or (1-2)(5-1)(-1-d)=0 da 1,5, -1 Eigen values are l-lolos bi for eigen vector, when d=-11 2 6 23 682=0 09 22-0 2x1+22-23=0 or 2x1=23 Kia et x1=1 then 23=2 so when d=1 1 600 4 482:0 0932=0 22-23-0→22:23 and ai is free -9 X3 SO K2 -o when d=5, B 61-0 22-421-23= 0 22-623 = 0 -> 2=6% so -42,7683-330 4211:523 o -6123 091 CS Scanned with CamScanner

X3=4 General Solution X(t)= cerit, let then 22 = 24 and x,=5 S So K3 = 94 4 5 Eigen vectors are Kia K2 K3 =24 2 4 det Kitce Kathe K3 ;] -f67 dat ul a(t) y(t) z(t) ceth teget 4 <3(4), 464)2(t))=//cie-+czet+sizest, 24032st 2cie-t+uczest 2 CS Scanned with CamScanner