Asymptotics of a Second-Order Differential Equation with a Small Parameter in the Case when the Reduced Equation Has Two Solutions

The boundary value problem for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative is examined. It is assumed that the reduced equation has two solutions with intersecting graphs. Near the intersection point, the asymptotic behavior of the solution to the original problem is fairly complex. A uniform asymptotic approximation to the solution that is accurate up to any prescribed power of the small parameter is constructed and justified.