DIFFERENTIAL EQUATIONS 821 



It will be seen from these examples that if one constant is 

 eliminated a differential equation of the first order is formed, 

 while the elimination of two constants produces a differential 

 equation of the second order. Therefore the solution of a dif- 

 ferential equation of the first order may contain one arbitrary 

 constant, while the solution of a differential equation of the second 

 order may contain two arbitrary constants. Confining our work 

 to differential equations of the first order, the two types which 

 occur most frequently in actual practice are those equations in 

 which the variables can be separated, and those equations which 

 can be solved by the use of an integrating factor. 



165. When the Variables can be Separated. Equations of this 

 type are such that all the terms involving x can be placed with dx 

 on one side, and all the terms involving y can be placed with dy 

 on the other side. Then one side can be integrated with respect 

 to x and the other side with respect to y. 



These equations have the form, or can be reduced to the form, 



Y +X =0 ....... (2) 



where X is a function of x and Y is a function of y, 



and |X dx = - (V dy + Const ..... (1) 



--+ Const ...... (2) 



(Lit 

 Example 1. Solve the equation x -j- = y + xy. 



Now x -- = y(x + 1) 



dy x 



and -j* - - = y 



dx x + I 



Const - 



J y 



x + log, x + c = log e y 



and y = xe y> xtP 



= Axe? where A = ef 



X 



