i dont know how to solve or use mathematica to solve this 2nd order ordinary differential equations?
You could write it as and solve it via power series. In Mathematica, use DSolve:
DSolve[y''[t]-k/(a-bt) y[t]==0,y,t]
which returns an answer in terms of Bessel functions. Might be interesting to reconcile that answer with the answer you get using power series.