184 MR. ROBERT RAWSON ON SINGULAR 



A gain j let 



f(iv) = cos iv, then f'(w) = — sin w ; 

 substitute these values in (21) and (22), then 



(dv dz \ 1 dw 



— + — -coswl- z-t- 

 ax ax / sin w ax , . 



c?a? /flfv d^ \ 1 dw ' ^' 



I -7- + -r cos ttf ) z -r~ 



\ay ay / sin w ay 



c + v + ^cos^ = o (36) 



Now w = o is not a solution of (35), since sini^ is zero, 

 and no value which can be given to (10) will satisfy the 

 equation sin iv= infinity. 

 Let 



••• c --^=° (38) 



is its complete primitive. This example is from Boole's 

 Diff. Eqs. Supp. vol. p. 17. 



In equation (34) put v = o, z=—logy, n=^—i, and 

 w = x; then # = is a solution which satisfies (37), but 

 which is not contained in (38) by giving a constant value 

 to (c), therefore it is a singular solution of (37). Boole 

 rejects the solution x=o because it does contain in its 

 expression (y). For what reason this rejection is made 

 does not appear. 



By reference to equations (31) and (32), put v = o, 



s= , and iv — y, then y = i is a solution of (37), but it 



is a particular solution by giving c = o in the primitive 

 y = e cx . 



Boole has obtained the solution y = o in two places, viz. 



