In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higher-order

QUESTION:

In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higher-order equations. We first outline the procedure for the third-order equation y___ + p1(t) y__ + p2(t) y_ + p3(t) y = 0. Let y1, y2, and y3 be solutions of this equation on an interval I . a. If W = W[y1, y2, y3], show that W_ = _______ y1 y2 y3 y_ 1 y_ 2 y_ 3 y___ 1 y___ 2 y___ 3 _______ . Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinants obtained by differentiating the first, second, and third rows, respectively. b. Substitute for y___ 1 , y___ 2 , and y___ 3 from the differential equation; multiply the first row by p3, multiply the second row by p2, and add these to the last row to obtain W_ = p1(t)W. c. Show that W[y1, y2, y3](t) = c exp__ p1(t)dt_ . It follows that W is either always zero or nowhere zero on I . d. Generalize this argument to the nth order equation y(n) + p1(t) y(n1) + + pn(t) y = 0 with solutions y1, . . . , yn. That is, establish Abels formula W[y1, . . . , yn](t) = c exp__ p1(t)dt_ (17) for this case.