Let y1 and y2 be two solutions of A(x)y″ + B(x)y′ C(x)y = 0 on an open interval I where A,...

Let y1 and y2 be two solutions of A(x)y″ + B(x)y′ C(x)y = 0 on an open interval I where A, B, and C are continuous and A(x) is never zero.

(a) Let W = W(y1, y2). Show that

Then substitute for Ay2″ and Ay1″ from the original differential equation to show that

(b) Solve this first-order equation to deduce Abel’s formula

where K is a constant.

(c) Why does Abel’s formula imply that the Wronskian W(y1, y2) is either zero everywhere or nonzero everywhere (as stated in Theorem 3)?