Before applying Eq. (19) with a given homogeneous second-order linear differential equation and a known solution y1(x), the equation must first be written in the form of (18) with leading coefficient 1 in order to correctly determine the coefficient function p(x). Frequently it is more convenient to simply substitute y = u(x)y1(x) in the given differential equation and then proceed directly to find v(x). Thus, starting with the readily verified solution y1(x) = x3 of the equation
x″y″ − 5xy′ + 9y = 0 (x > 0),
substitute y = vx3 and deduce that xv″ + v′ = 0. Thence solve for v(x) = C ln x, and thereby obtain (with C = 1) the second solution y2(x) = x3 ln x.
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