178 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



The first integral is: 



where, 



Q = Pn, 



o p dPn 



Qn-l = Pn-l ~ -, 



dPn-l . 



P 



If the first integral is an exact differential equation the process may be con- 

 tinued as long as the coefficients of each successive integral satisfy the condition 

 of integrability. 



8.311 Non-linear differential equations. A non-linear differential equation of 

 the nth order: 



\dx n ' dx n ~ v ' ' dx' ' 



to be exact must contain -j=2 in the first degree only. Put 



d n ~ l y _ d n y = dp < 

 dx n ~ l dx n dx 



Integrate the equation on the assumption that p is the only variable and 



~ its differential coefficient. Let the result be FI. In F dx dVi, ^ is 



dx dx n ~ l 



the highest differential coefficient and it occurs in the first degree only. Repeat 

 this process as often as may be necessary and the first integral of the exact dif- 

 ferential equation will be 



If this process breaks down owing to the appearance of the highest differential 

 coefficient in a higher degree than the first the given differential equation was 

 not exact. 



