Asymptotic solutions for second-order differential equations (Q1910388)

The asymptotic form of two linearly independent solutions of the second-order differential equation \((p(x) y'(x))'+ r(x) y'(x)- q(x) y(x)= 0\) as \(x\to \infty\) is studied for the case when \(r\) is small compared to \(p\) and \(q\) in the sense that \[ {r\over (pq)^{1/2}}\to \infty\quad \text{as} \quad x\to \infty. \] The results are illustated with examples.