On Rational Spline Solutions of Second-Order Linear Differential Equations with an Unbounded Second Coefficient

For a differential equation $$u'' + P(x) u' + Q(x) u=H(x)$$ , $$0< x<1$$ , with an unbounded coefficient $$P(x)$$ in a neighborhood of zero, the question of approximately solving two-point and one-point boundary value problems using rational spline functions is studied by passing to an equation of the form x² y'' + x p(x) y' + q(x) y=f(x) , analogous to the equation considered previously by the authors. In this case, the two-point problem is studied taking into account the behavior of the modulus of continuity of its solution, and the one-point problem is studied taking into account the order of growth of its solution in a neighborhood of a singular point.