Find the values of alpha such that the system xt 3 9 alpha

Find the values of alpha such that the system x(t) = (3 -9 alpha -3) x(t) will have periodic solutions.

Solution

the given eqn can be written as

X \' = 3x - 9y can be written as ( D - 3) x +9y =0 ----(1)

y \' = a x - 3 y ------ - ax + ( D +3) y =0 -----(2)multiply eqn 1 by (D+3) and 2 by 9 (subtract)

(D+3) (D-3) x + 9ax =0 => ( D² - 9 +9a) x=0 the solution is aperiodic fn if

- 9 +9a = k²  => 9 ( a -1) = perfect square

1. hence the possible values of \' a\' are a =2 => 9 (2-1 ) =9 =k²  ( a =2 )

2 . a-1 = 9 => a= 10 ANS when a= 2 , 10 the solution is periodic