I think you should apply D'Alambert method to find a linear independent solution (from that you already found) of the homogeneous.
Then when you'll have y1h=-ln(x) and y2h=found using D'Alambert you can apply the Lagrange method to find the complete solution.
Sorry, It's have copyright but if you try from basic,it's may be advanced. Try computer programming generate solution. The number of control point may in form 2N , I'm think should try sample by niquist rate .