Ogilvie 



dynamics problems. It will be necessary to discuss this point further 

 at an appropriate place. 



A related question concerns the precise definition of the small 

 parameters that we use to formulate the approximate problems. In 

 this paper I avoid defining the small parameter quantitatively. It Is 

 usually unnecessary and It Is dangerous. I shall return to this 

 point also. 



1.2. Matched Asymptotic Expansions 



For most of our problems , the approach advocated by 

 Van Dyke [ 1964] Is entirely adequate. I shall assume that the reader 

 Is familiar with (or has access to) Van Dyke's book. Only a few 

 definitions and concepts will be mentioned here. 



Perhaps the simplest problem that demonstrates the applica- 

 bility of the method of matched asymptotic expansions Is the following: 

 Find the solution of the differential equation, 



ey + 2y + y = , 



subject to the Initial conditions: 



y(0) = 1; y(0) = 0. 



The parameter € Is to be considered snnall, and. In fact, we want 

 to know how the solution of this problem behaves as c -* 0. Now, 

 If we set c = 0, the order of the differential equation Is reduced, and 

 two Initial conditions cannot be satisfied. Therefore, one cannot 

 obtain a series expansion for the solution by a simple Iteration scheme 

 which starts with the solution for the limit case, e = 0. 



The exact solution for this problem Is: 



P,t Pjt 



y(t) = --22l iPiJL— , 



where 



_-i-7rr 



P| = c 



€ 



- 2/6 J 



^^-.zAJLfH ^ . ,/z. 



670 



