the problem's original differential equation (shown by substituting it and its derivatives into the original equation). Also, the boundary conditions produced by this final solution match the values given in the problem, up to a constant multiple. This implies, due to the uniqueness of the solution, that the matched asymptotic solution is identical to the exact solution up to a constant multiple. This is not necessarily always the case, any remaining terms should go to zero uniformly as . Not only does our solution successfully approximately solve the problem at hand, it closely approximates the problem's exact solution. It happens that this particular problem is easily found to have exact solution which has the same form as the approximate solution, by the multiplying constant. The approximate solution is the first term in a binomial expansion of the exact solution in powers of . Location of boundary layer Conveniently, we can see that the boundary layer, where and are large, is near , as we supposed earlier. If we had supposed it to be at the other endpoint and proceeded by making the rescaling , we would have found it impossible to satisfy the resulting matching condition. For many problems, this kind of trial and error is the only way to determine the true location of the boundary layer. Harder problems The problem above is a simple example because it is a single equation with only one dependent variable, and there is one boundary layer in the solution. Harder problems may contain several co-dependent variables in a system of several equations, and/or with several boundary and/or interior layers in the solution. It is often desirable to find more terms in the asymptotic expansions of both the outer and the inner solutions. The appropriate form of these expansions is not always clear: while a power-series expansion in may work, sometimes the appropriate form involves fractional powers of , functions such as , et cetera. As in the above example, we will obtain outer and inner expansions with some coefficients which must be determined by matching. Second-order differential equations Schrödinger-like second-order differential equations A method of matched asymptotic expansions - with matching of solutions in the common domain of validity - has been developed and used extensively by Dingle and Müller-Kirsten for the derivation of asymptotic expansions of the solutions and characteristic numbers (band boundaries) of Schrödinger-like second-order differential equations with periodic potentials