Solve Boundary Layer ODEs (AsymptoticDSolveValue)

This example shows a perturbation problem for a linear differential equation in which the order of the equation decreases when the perturbation parameter is set to 0. For a fixed small value of the parameter, the nature of the solution depends on the relative scales for and . Hence, the solution can be regarded as being composed of a boundary layer near the left endpoint, 0, where is much larger than , and an outer region near the right endpoint, 1, where is much larger than . Such problems are called singular perturbation problems or boundary layer problems.

Obtain the leading term WKB approximation for a boundary layer problem.

Compare with the exact solution.

Plot the solution and the approximation.