All-101 100 



The solution to the first of the tv;o equations is given 

 by (*+). Since the complete solution will only be approximate, 

 in that terms of order e have already been neglected, ■^■^^ need 

 only be determined approximately. 



It is suggested by the above considerations that v/c 

 find a formal method for writing our eqiiation so that the 

 magnitudes of the terms arc expressed by the coefficients and 

 that the derivatives, etc., be of order unity. ¥q can do this 

 by stretching the x coordinate near the boundary i.e., by de- 

 fining a new x coordinate so that a particular distance in x 

 becomes a much larger distance in the nev/ coordinate. 



Formally, vc operate as follows. Let x be replaced by 

 the coordinate ^ such that 



X = e^C 



where n is to be determined. Then the equation (5) becomes 



In choosing n we note that it must be positive if the 

 X coordinate is to be stretched. Thus of the terms which 

 originally had coefficient e, e~™+^ '^hffll ^^ ^^^ largest 

 since it has the largest coefficient (n.b. \)j^r, '^brrrr' 

 ^brr ? '^-'b ^•'^® ^^^ same order of m.agnitude). This term 

 is m.atGhed with e"^ i{j, r, the rem^alning large term in the 

 differential equation, and by equating the coefficients of the 

 above two terms, v/e have n = 1/3. 



