DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 23 



Substituting- these expressions in the transformed auxiliary equation, 

 it becomes do) = , wlience u = k ^ that is 



The former of the auxiliary equations now gives 



y — Iw = Ä . 



Now by putting- k = /(*) and eliminating k and ct, we might obtain 

 the general first integral. It will, however, be impossible to integrate, 

 and, therefore, we must instead of the general first integral use the com- 

 plete one. Therefore, supposing /(«.) to be an arbitrary constant k^ the 

 complete first integral to be integrated becomes 



by the integration of which we get :,„ + ^''o.i = '^{y — ^^^) i ^^^^ ^Y ^ 

 new integration the mixed primitive 



c = Cp (y — k a-) 4- œ -^{y — kx) . 



Ex. 3. To exemplify the method of limits, that may be used, when 

 (20) has equal roots, we take the simple equation x':.jj,-\-2,vy:,i+y-:„.2 =0, 

 which, however, may more easily be integrated by the usual method of 

 Monge. We here integrate the more general equatioii 



x% , + ai2y + k) .-,,1 4- y(yrk):o, = 0, 



from Avhich the given equation will be obtained by putting /.; = 0. 



The equation (20) gives for the latter equation the roots m^ = 

 y 1 y-T-k 



«^1.0 + 3/~o,i = Ä + -s^ 



and »)?., = tiX_ . Hence the two first integrals will be 



X ' X 



X. 



Now eliminating Cj „ and ^o i between these equations and 



dz = ,-10 dx 4-^0.1 dy , 



we get 



kd: = k: 



^ + (y+^dx-dy) ^ r-'à±\ + [dy - idx) Ç p), 



X \ X I \ X I \ X / \xJ 



