Question

Differential Equations

Find a general solution of the system x'(t)=Ax(t) for the given matrix A. 8 13 5 -8 x(t) (Use parentheses to clearly denote the argument of each function.)

Answer 1

A = [8 5 - 13 - 8] Let x' (t) = [x' y'] = > x (t) = [x y] Characteristic equation lambda^2 - 0 lambda + 1 = 0 or lambda^2 = -1 lambda = plusminus i [A - i I] x_1 = 0 = > [8 - i - 13 5 - 8 - i] [x_1 x_2] = [0 0] x_1 = [8 + i 5] = > 5 x_1 - (8 + i) x_2 = 0 [A - (- i) I] x_2 = 0 = > [8 + i 5 - 13 - 8 + i] [x_1 x_2] = [0 0] = > 5 x_1 + (- 8 + i) x_2 = 0 x_2 = [8 - i 5] The general solution is given by x (t) = A x_1 e^i t + B x_2 e^- i t A & B are constants = > [x y] = A [8 + i 5] e^i t + B [8 - i 5]