This problem deals with generalization of Abel's theorem to higher order equations. Consider rst the third order ODE y000 + p1(t)y00 + p2(t)y0 + p3(t)y = 0: (1) Let y1(t), y2(t) and y3(t) be solutions of this equation on an open interval I where p1(t), p2(t) and p3(t) are continuous. (a) If W(t) is the Wronskian for y1, y2 and y3, show that dW dt = y1 y2 y3 y01 y02 y03 y000 1 y000 2 y000 3 (2) using the fact that the derivative of a 3-by-3 determinant is the sum of three determinants obtained by di erentiating the rst, second and third rows, respectively.
(b) Use (1) to nd y000 1 , y000 2 and y000 3 in terms of y00, y0 and y and substitute these expressions into the last row of the determinant in (2). Add the sum of the rst row multiplied by p3 and the second row multiplied by p2 to the last row (note that this operation does not change the determinant) to obtain dW dt = p1(t)W
(c) Show that W(t) = C exp Z p1(t)dt ; (3) where C is a constant. Explain why it follows that W is either always zero or nowhere zero on I.
(d) Generalize this argument to the nth order ODE y(n) + p1(t)y(n1) + + pn(t)y = 0 with solutions y1; : : : ; yn by showing that (3) holds for this case.
(e) Use (3) to nd the Wronskian of a fundamental set of solutions of ty000 + 2y00 y0 + ty = 0; t > 0: