Question

6. (3 -10 Consider the system = AX where A = . The matrix A has eigenvalues dt 12 -5 ) 2 = -1+2i. Find the general solution of this system. (10 pts)

Answer 1

Date Page Given that the system dx = AX at : where A= 13 10 ( 2 -5 and I has eigenvalues d = 1tzi Find eigen vectors d=-112 2 27) 4-0 (3-1-1+2i) -10 - 2 -5-(-1728)/ 15- 2 ľ -10 114 1 2 -A-21 TY) = (5-2i) y - 10% =0) - 2y + (-4-21) Y2=0 srt 24 - (utrů ) Y ₂ = 0 Iniz y = (2+ ( ) Y₂ (2 + 1 ) Y 1-42= 1 then y= 12th So, the ) eigen vector is 1 2 = -1-2 1 A-Il-ou 3+ (1+zi) -10 2 -5+0+ ) ) Stri -10 11y = 0 2 -4+21 1 sonth 7 24 + (-4+21) 42 = 0 y = (2-1 / 42 so the eigenvector is = /2-1] genvector is = 3-1 if y=t- very Teacher's Signature.....

Date Page The solution that we get value and eigenvector from the first eigen I-1+2it2til xilt) = e 2417) = (cosat tisinat) (ati - = 2 et (cosatt i sinat) tie t ( Cosatt isinat et (Losat tisinato x(x) = /2 et flosat et osat + i lepinatt/et loszt xolt) = /2 et cosat - etsinat ti /petsinst tēt cosat 1 ēt Losat I ét sinat xy(t) = ult) ti olt) - ult) & vlt) are two Loi soln of the system of D: E. Hence the general solution to a system with complex root X(t) = cult) +Rult) for our system the general sol is x lt) = 2 et Cost - ēteint talretsinzt te losst et coset etsint