Suppose that one solution y1 (a ) of the homogeneous second-order linear differential equation y" + p(x)y' + q(x)y=0 (1) is known (on an interval I where p and q are continuous functions). The method of reduction of order consists of substituting y2 (a) = v(ac) y1 (a) in (1) and attempting to determine the function v(a ) so that y2 ( ) is a second linearly independent solution of (1). After substituting y = v(x) y1 (a ), use the fact that y1 ( ) is a solution to deduce that yiv" + (2y1 + pyl)u' = 0. (2) If y1 (a) is known, then (2) is a separable equation that is readily solved for the derivative of v' (a) of v(a). Integration of v (a) then gives the desired (non constant) function of v(a). Thus, starting with the readily verified solution y1 (ac) = a of the equation x2 y' - 5xy' + 9y =0 (x > 0), substitute y = va' and deduce that av" + v' = 0. Thence solve for v(a) = CIn x, and thereby obtain (with C = 1) the second solution y2 (a ) = 23 In x