1 66 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



8.045 The equation does not contain y: 



f(x, p) = o. 

 It may be solved for p, giving, 



which can be integrated. 



It may be solved for x, giving, 



x = F(p), 

 which may be solved by 8.043. 



8.050 Equations homogeneous in x and y. 

 General form: 



(a) Solve for p and proceed as in 8.001 



(b) Solve for 2 ; 



x 



y = *f(p). 



Differentiate with respect to x: 



dx _ f'(p)dp 



x ~ p-/(py 



which may be integrated. 



8.060 Clairaut's differential equation: 



1. y = P* 



the solution is: 



y = cx+f(c). 



The singular solution is obtained by eliminating p between (i) and 



2. x+f'(p)=o. 



8.061 The equation 



y = xf(p) 

 The solution is that of the linear equation of the first order: 



to f'(P) 4>'(P) 



dp P-M P-f(P)' 



which may be solved by 8.002. Eliminating p between (i) and the solution of 

 (2) gives the solution of the given equation. 



