All-101 98 



i^x "= ~ (1+a sin t) sin nsy (3) 



Thus, a possible solution is 

 ijjj_ ^ (i+a sin-r) sin nsy [- x + C]_x(y5'^) 1. (^) 



Wg arc now faced with a dilomna, howovcr. ijj as given 

 in (^1-) provides one arbitrary function of y and 1 to satisfy 

 the four conditions on the boundaries x = 0, x = r. If our 

 assumption that \)j is everywhere a smooth function is correct, 

 then we are at a loss to find a complete answer to the problem. 

 For if \jj and its derivatives have the same order of magnitude 

 everywhere, the only possible solution is of the form 

 ^. + 0(e) and it is not possible to satisfy all boundary con- 

 ditions. 



It is obvious, therefore, that ^l) cannot be smooth 



everywhere. In particular, in order for the full solution 



to be different fromilJj_ + 0(e), at least one of the terms, 



lb , ii) , or ^iJ must b: of order e"-'- in some part of 

 ^xxxx' ^xxyy' YYYY 



the domain under consideration so that the approxiiiiation of 



neglecting terms of order e will not reduce the order of the 



differential equation. If i|j is smooth away from the boundaries 



and if derivatives with respect to x are large, so that 



lb is of order e~l, near x = 0,r, then the problem is one 

 ^xxxx ' 



of the boundary layer type. ¥e shall proceed formally on the 

 assumption that this is true, realizing that if it is not the 

 case, wo shall be led to a contradiction. 



