The uniqueness theorem dictates that given the initial condition x(5) = 0 for the differential equation x'(t) = Ax, the solution x∗(t) must be 0 for all t, due to the system being linear and homogeneous.
- The question addresses the solution of a system of differential equations x'(t) = Ax, where A is an n × n matrix of real numbers.
- Given that the initial condition is x(5) = 0, the uniqueness theorem for differential equations implies that the solution to the initial value problem is uniquely determined by its initial condition.
- Since x(5) = 0 and the system is linear and homogeneous (no constant or non-homogeneous terms), it follows that x∗(t) = 0 for all t.
- More informally, if you start a system at a stationary point (where all derivatives are zero), and there are no external forces to move it away from that point, then the system will remain at the stationary point for all time.
- Hence, x∗(t) stays zero for all t.