180 MR. ROBERT RAWS ON ON SINGULAR 



by ordinary differentiation, and the term singular solution 

 of (21) can be justified only on the ground that it serves 

 admirably to distinguish the two kinds of solution referred 

 to in (22) and (23). 



In the general theorems when a solution is said to be 

 singular, it is meant to be so only when it is not contained 

 in the complete primitive by giving a constant value to (c) 

 independent of x. (See Boole's Diff. Eqs. p. 163, 2nd 

 ed.) 



8. The following examples will illustrate the general 

 f ormulae : — 



Let 



(2) 2 - 2v/ ^£ +2V ^ =i - ' (24) 



Required the singular solution, if any. 

 This equation admits of the form 



The condition of equal roots is evidently 



Vx + ij = i (26) 



But this is not a solution of (24), therefore the principle 



of equal roots fails to give the singular solution. 



Again, 



dp 1 



dy */x + y 



= infinity 3 if x-\-y = o, 



And this is the singular solution required. 

 The complete primitive of (24) is 



(c— x) 2 -(y + A/x + y)(c — x)+y</x + y=o. . (27) 

 The condition of equal roots is 



*+y=y z , (28) 



