We look for a solution to the entire equation of the form . Then . There are, in fact, an infinite number of such functions, u(x) and v(x), (given a solution y, pick any function to be u(x) and solve for v), we can simplify, and narrow the search, by requiring that . That reduces y' to so and the equation becomes
Our clever requirement that means that there are no u" or v" terms in the equation and the fact that y1 and y2 satisfy the homogeneous equation means that there are no u or v terms! We have, now, two equations, and to solve for u' and v'. For the first, we can divide through by x-1 to get . Solve those equations, algebraically, for u' and v', then integrate to find u and v.