Exercise 20.  Many texts give an alternative definition for  ,  starting with (5-1) as the definition for  .  

20 (c).  Using a similar procedure for  ,  show we wind up getting a pair of solutions

,  and    where and are functions of y alone.

Solution 20 (c).

See text and/or instructor's solution manual.

Solution.   The result in part (a) implies that  .

This means    is constant with respect to x, so  ,  where is a function of y alone.

Details.   From part (a)   ,  hence

                       

This means    is constant with respect to x, so  ,  where is a function of y alone.

Combining this with the result of part (b) we see that

                    ,    and  

                    ,

where and are functions of y alone.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell