Before applying Eq. (19) with a given homogeneous
Chapter , Problem 37(choose chapter or problem)
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 D v.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/ D x3 of the equation x2y00 5xy0 C 9y D 0 .x > 0/; substitute y D vx3 and deduce that xv00 C v0 D 0. Thence solve for v.x/ D C ln x, and thereby obtain (with C D 1) the second solution y2.x/ D x3 ln x.
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