9. Let x...,x) be linearly independent solutions of x' = P(t)x, where P is continuous on...

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9. Let x...,x) be linearly independent solutions of x' = P(t)x, where P is continuous on %3D (a) Show that any solution x = z(t) can be written in the form z(1) = cx"(t) +..+ C,x" (1) %3D for suitable constants c1,..., Cy. Hint: Use the result of Problem 12 of Section 7.3, and also Problem 8 above. (b) Show that the expression for the solution z(1) in part (a) is unique; that is, if z(1) = kix"(1) +...+ k,x" (1), then ki Hint: Show that (k - c)x(1) +. the linear independence of x = C1.... Ka = Cn. ...+ (k, - c,)x" (t) = 0 for each i in a <t < B, and use 9. Let x...,x) be linearly independent solutions of x' = P(t)x, where P is continuous on %3D (a) Show that any solution x = z(t) can be written in the form z(1) = cx"(t) +..+ C,x" (1) %3D for suitable constants c1,..., Cy. Hint: Use the result of Problem 12 of Section 7.3, and also Problem 8 above. (b) Show that the expression for the solution z(1) in part (a) is unique; that is, if z(1) = kix"(1) +...+ k,x" (1), then ki Hint: Show that (k - c)x(1) +. the linear independence of x = C1.... Ka = Cn. ...+ (k, - c,)x" (t) = 0 for each i in a <t < B, and use