Reduction of Order: Suppose you know one solution u1(x) to the second order homogeneous differential equation u" + a(x)u′ + b(x)u = 0.
(a) Show that if u(x) = v(x) u1(x) is any other solution, then w(x) = v'(x) satisfies a first order differential equation.
(b) Use reduction of order to find the general solution to the following equations, based on the indicated solution:
(i) u" - 2u' + u = 0, u1(x) = ex
(ii) xu" + (x - 1)u' - u = 0, u1 (x) = a - 1
(iii) u" + 4xu' + (4x2 + 2) u = 0, u1(x) = e-x2
(iv) u" - (x2 + 1) u = 0, u1(x) = ex2/2