Question

Let

A = ( a₁ 0 ... 0

0 a_{2 ... 0}

... ... ...

0 0 a_n)

be an n * n matrix, where a1, a2, . . . , a_n are nonzero real numbers.

(a) Find the general solution to the system of equations -> ->

x'  = A * x
(b) Solve the initial value problem x₁(0) = x₂(0) = · · · = x_n(0) = k, for some constant k.

(c) Solve the initial value problem (x₁(0)  x₂(0) · · · x_n(0)) = ( 1 2 .... n).

Let 0 a1 0 0 0 A2 A = ... ... ... 0 0 An be an n x n matrix, where 21, A2, ..., An are nonzero real numbers. = Aš = (a) Find the general solution to the sysem of equations z (b) Solve the initial value problem x1(0) = x2(0) Xn (0) = k, for some constant k. X1(0) X2(0) 2 (c) Solve the initial value problem : 2n (0) n

Answer 1

we First determine the eigenvalues and correspon - ding eigenrectors eigenrectors of t. * is diagonal matrin whose diagonal elements a are a, a where ai (i=1,2,.. be non- zero real numbers. tence the eigenvalues of a are au, ao 2 an we then know that if t be diagonal elements for the Koth diagonal element of car eigenvector be OO-.-YO...O > kth term Hence the eigenvectors of A corresponding to the eigenvalues assas, 2 an are () OO-..0- respectively 9 Then the general solution of solution of 2 = An be i = a put x = + Geart (1) t...tene Where agli In are arbitrary constants 7 че eart q eart ny = (1) cne eant

( b) b from (1) We get a eart (+) M (*) vu. eat (2) an (*) an eant given that 210) = m (0) un(o)=k Putting t 0 in (2) , we get (0) " (0) q e - anco) che CA TA K k کی ... این c k Comparing, we get Cn=k. G=k, e =k, in (2), we get Putting the values 74 () keaut ke at az (A) u = home) o ant rin (A) @) Gwen that, 2 M(0) mo) حیا 1. n un (0) in (2) We get 2 Putting *20 74(0) R2(0) sin(e) qe ine

G 2 IM here - Where 2 it is obtained using (3) n Cn Comparing, we get q= 1, 5= 2, cn =n an 2 Hence the solution from (2) we get eart zegt I = 24 (1) M(*) in (A) . neant

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