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 form z(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, if z(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 use the linear independence of x(1) , ... , x(n) .