Question

Use the elimination method to find a general solution for the given linear system, where the differentiation is with respect to t. x' = 6x – y y' = 6y – 9x

Answer 1

solution: Given, x' = 6x-y y' = 64 - gx That is, dx = 6x-4 =by-gx he From we get, 9x = 64 - dy dt by- dy [by - du From and (A we get, 6 a [by - dy 2 [by - duy 6 oktober dy ] 6 [69 - dy ] - y - dy ] - gy 364 - 6 du - gy 6 [ by ニ AP E d dt2 -12 dy + 27y 0 = dt (5 is, dt - Now, Auxiliary equation of m² - 12m +27=0 [m = d m²-3m-om +27= 0 > m(m-3) -g(m-3) = 0 → (M-9) (m-3) = 0 > m=9, 3 3t Hence, y(t) = q e t ce where I. (2 are arbitrary constant gt

Y(t) = Ciegt + c2e3t From we get, dy dt 9x = 6 - 9x= >9x= -34 e Hence, X(t) = 3 9x= 64 (9 est + C2 e3t 6( e*t + (2 e²t) - & (6G est +6 (263t) - (99 68t + 3c2e3t) 9t + 3 C2 e 3t X = - 3 [-9e9t + C2 e 3t t] where a, (2 arbitrary constant 1/3 [-q edt + (2 e3t ] where G, C2 are arbitrary constants.