DIFFERENTIAL EQUATIONS OF THE n"' ORDER. 31 



and this being- integrated r times with respect to y gives 



where it is to be observed, that f Ç,(y)dy is itself an arbitrary function 

 as well as (p, (y) and , therefore , may simply be changed into x, iy) • 



The last equation may evidently be written 



i—n-(j-r 

 1=0 



the accordance in form with (48) is evident. 



We might also have transformed (53) by putting = ^ = -' , 

 whence s„_,, = <_p_,,,-.^ ; 



this equation would then have assumed the form 



or S C,^r='„^Q-r-i.i = F{x,y). 



i=fl 



Now if the general solution of the last equation be found, and we 

 suppose it be z' = (p(x,y), we have only to seek the general solution 

 from the equation 



which is immediately integrable. 



Corollary V. If F = F{x,y) be a rational integral polynôme, we 

 may, by its differentiation with respect to one or both of the independent 

 variables, form a new linear differential equation with constant coefficients 

 and with a right member either zero or of one of the forms considered in 

 foregoing corollaries. After the integration of this new equation its solu- 

 tion must be so restricted , that it satisfies the given equation. Of the new 

 equation we must, of coursp, use the general solution, in order to be sure, 

 that the general solution of it may contain the general solution of the given 

 equation as a special case. 



The following examples will fully illustrate this method. 



