Asymptotic formulae of Liouville-Green type for a general fourth-order differential equation (Q1293499)

The author considers the following equation \[ (p_0y'')''+ (p_1 y')'+1/2 \sum^1_{j=0} \bigl[\{q_{2-j} y^{(j)}\}^{(j+1)} +\{q_{2-j} y^{(j+ 1)} \}^{(j)} \bigr]-p_2y=0. \] The functions \(p_j\), \(j=0,1,2\), and \(q_j\), \(j=1,2\), are defined on \([a,\infty)\) and are not necessarily real-valued. He proves that the above equation has the solutions \(y_k(t)\), \(k=1,2,3,4\), such that \[ y_k(t)\sim p_0^{ -{1\over 8}}(x)p_2^{-{3\over 8}}(x)\exp \left(\int^x_a \lambda_k(t)dt \right),\;x\to\infty \] where \(\lambda_k(t)\) are characteristic roots.