DIFFERENTIAL EQUATIONS 165 



8.041 The equation can be solved as an algebraic equation in p. It can be 

 written 



(p-R 1 )(p-R 2 ) (p-R n )=o. 



The differential equations: 



P = Ri(*, y), 

 p = R z (x,y), 



may be solved by the previous methods. Write the solutions: 

 fi(x, y, c) = o; f 2 (x, y, c) = o; 



where c is the same arbitrary constant in each. The solution of the given 

 differential equation is: 



fi(%, y> c)f*(x, y,c) fn(x, y, c) = o. 



8.042 The equation can be solved for y: 

 Differentiate with respect to x: 



It may be possible to integrate (2) regarded as an equation in the two variables 

 x, p, giving a solution 



3(D(T 1) C) == O. 

 T\**J ) / 



If p is eliminated between (i) and (3) the result will be the solution of the given 

 equation. 



8.043 The equation can be solved for x: 



Differentiate with respect to y: 



2. - = 

 If a solution of (2) can be found: 



3. ' Cv> P) c } = - 



Eliminate p between (i) and (3) and the result will be the solution of the given 

 equation. 



8.044 The equation does not contain x: 

 It may be solved for p, giving, 



which can be integrated. 



