Equations by Factors and Differentiation. 389 



the highest order of jt?, q, &c. not exceeding the highest order of 



the differential coefficients in F. Suppose also that this equation 



is satisfied if F = 0, which equation involves, as is known, the 



dY d 2 F 

 consequences — = 0, -7-^ = 0, &c. Then any integral of the 



equation yfr — which is reducible to an order lower than the 

 order of F = is to be regarded as an integral of the latter equa- 

 tion. Such integration of a given equation F = is in principle 

 an extension of the method of integrating by factors, or of that 

 by differentiation, or of a combination of both operations. 



Clairaut's form, y=p%+f{p), furnishes an instance of inte- 

 grating by differentiation only. By differentiating we have 

 q(l+f'(p)^)=0, which equation is satisfied if # = 0, and, by 

 consequence, if p—c and y = ca? + c ! . By substituting c fovp in 

 the given equation, it will be seen that c f — /(c), so that the proper 

 form of the integral is y = cx+f{c). On eliminating c between 

 this and the derived equation p = c, the original equation is re- 

 produced. 



For exemplifying integration by a factor let us take the 

 equation 



J_ yz__=o. 



Multiplying by the factor p we obtain 



p _ vn = d- y ^ n . 



whence by integration y=c Vl+p* 2 , an equation of a lower 

 order by one unit than that of the proposed equation, and con- 

 sequently a first integral of the latter. A second integration 

 gives 



^ + c / = clog e | + ^/^ + l = clog e (^+ vT+p), 



the known equation of a catenary. 



If the proposed equation be 1 +p 2 —qy=0, it would be made 



intestable by the factor — — r, and the same result as before 



would be obtained. In fact, differential equations differing in 

 form are one and the same if the factors which make them inte- 

 grate reduce them to the same form ; but no general rule exists 

 for finding the appropriate factor in each case. The foregoing 

 instances come under the general theorem above enunciated, 



because, if we put F for — =^ — — », the equation 



