2. In this problem we outline a proof of Theorem 7.4.3 in the case n=2. Let x(1) and x(2) b ...
2. 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 α<t<β, and let W be the Wronskian of x(1) and x(2). (a) Show that dtdW=∣∣dtdx1(1)x2(1)dtdx1(2)x2(2)∣∣+∣∣x1(1)dtdx2(1)x1(2)dtdx2(2)∣∣.
(b) Using Eq. (3), show that dtdW=(p11+p22)W (c) Find W(t) by solving the differential equation obtained in part (b). Use this expression to obtain the conclusion stated in Theorem 7.4.3.
To prove the given equality, we'll start by expanding the derivative of the Wronskian, dW/dt, using the definition of the Wronskian and the product rule for derivatives.The Wronskian of two functions x? and x? is defined as: