Let x(1), ... , x(n) be linearly independent solutions of x = P(t)x, where P is

Chapter 7, Problem 9

(choose chapter or problem)

Let x(1), ... , x(n) be linearly independent solutions of x = P(t)x, where P is continuous on < t < .(a) Show that any solution x = z(t) can be written in the formz(t) = c1x(1)(t) ++ cnx(n)(t)for suitable constants c1, ... , cn.Hint: Use the result of of Section 7.3, and also above.(b) Show that the expression for the solution z(t) in part (a) is unique; that is, ifz(t) = k1x(1)(t) ++ knx(n)(t), then k1 = c1, ... , kn = cn.Hint: Show that (k1 c1)x(1)(t) ++ (kn cn)x(n)(t) = 0 for each t in < t < , and usethe linear independence of x(1), ... , x(n).