176 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



Put 



d n -*y _ v 

 dx n ~ 2 



- F(Y) 

 dx 2 



which may be solved by 8.301. If the solution can be expressed: 



Y = <t>(x), 



n 2 integrations will solve the given differential equation. 

 Or putting 



/dY C dY r YdY 



\ci + \i/(Y)r*l {ci + \i/(Y)\* / 

 *X .X 



where the integration is to be carried out from right to left and an arbitrary 

 constant added after each integration. The solution of the given differential 

 equation is obtained by elimination between this result and 



F = 4>(x). 



8.304 Differential equations of the second order in which the independent 

 variable does not appear. General type: 



pt dy &y\ _ 



Put 



P = dx' P dy = dx 2 ' 

 A differential equation of the first order results: 



If the solution of this equation is: 



*'-/M, 



the solution of the given equation is, 



8.305 Differential equations of the second order in which the dependent variable 

 does not appear. General type: 



dy d 2 y^ 



Put 



. _dy dp _ d 2 y 

 P ~ dx' dx ~ dx 2 ' 



