In this problem we outline a proof of Theorem 7.4.3 in the case n = 2. Let x(1) and x(2)be solutions of Eq. (3) for < t < , and let W be the Wronskian of x(1) and x(2).(a) Show thatdWdt =dx(1)1dtdx(2)1dtx(1)2 x(2)2+x(1)1 x(2)1dx(1)2dtdx(2)2dt.(b) Using Eq. (3), show thatdWdt = (p11 + p22)W.(c) Find W(t) by solving the differential equation obtained in part (b). Use this expressionto obtain the conclusion stated in Theorem 7.4.3.(d) Prove Theorem 7.4.3 for an arbitrary value of n by generalizing the procedure of parts(a), (b), and (c).

Lecture 18: The Chain Rule (Section 3.7) √ Consider the composite function h(x)= x2 +2 x − 3. How to diﬀerentiate ▯ 2 2 To get an idea, ﬁnHd (x)fi H(x)=( x +2 x − 3) .