I am currently studying a course in Asymptotic and Perturbation Methods and we have recently started discussing "Boundary Layer problems". It is not clear to me, however, exactly what form "Boundary Layer problems" typically take. In class, we have calculated approximate solutions for second order differential equations, where a small parameter multiplies the highest order derivative in the equation and the solution is subject to two boundary conditions. Why though do the equations I am presented with have to be solvable? Why does there have to be a a solution for any value of the small parameter and, further, why does the solution have to suffer a rapid change at some point? What other situations are classed as "boundary layer problems"?
Thank you in anticipation of your help.