In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higherorder

Chapter 4, Problem 20

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In this problem we show how to generalize Theorem 3.2.7 (Abels theorem) to higherorder equations. We first outline the procedure for the third order equationy + 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 thatW =y1 y2 y3y1 y2 y3y1 y2 y3.Hint: The derivative of a 3-by-3 determinant is the sum of three 3-by-3 determinantsobtained by differentiating the first, second, and third rows, respectively.(b) Substitute for y1 , y2 , and y3 from the differential equation; multiply the first row byp3, multiply the second row by p2, and add these to the last row to obtainW = p1(t)W.(c) Show thatW(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 equationy(n) + p1(t)y(n1) ++ pn(t)y = 0with solutions y1, ... , yn. That is, establish Abels formulaW(y1, ... , yn)(t) = c exp p1(t) dtfor this case.