Use variation of parameters to solve the given system x 1 6

Use variation of parameters to solve the given system. x\' = (1 6 -1/6 1)x + (csc t sec t) e^t. Recall from (14) in Section 8.3 that x = Phi(t) Phi^-1 (t_0) x_0 + Phi(t) integral_t_0^t Phi^-1 (s) F(s) ds solves the initial value problem x\' = AX + F(t), X(t_0) = x_0 whenever Phi(t) is a fundamental matrix of the associated homogeneous system. Use the above to solve the given initial-value problem. x\' = (5 3 3 5)x + (6e^8t 6e^2t), x(0) = (1 1)

Solution