DIFFERENTIAL EQUATIONS 177 



A differential equation of the first order results: 



If the solution of this equation is: 



?-/(*), 



the solution of the given equation is: 



y = c, + f}(x}dx. 



8.306 Equations of an order higher than the second in which either the inde- 

 pendent or the dependent variable does not appear. The substitution: 



dy * 

 dx = p > 



as in 8.304 and 8.305 will result in an equation of an order less by unity than the 

 given equation. 



8.307 Homogeneous differential equations. If y is assumed to be of dimensions 

 n, x of dimensions i, -~ of dimensions (n - i), -p; of dimensions (n - 2), 



..... then if every term has the same dimensions the equation is homogeneous. 

 If the independent variable is changed to 6 and the dependent variable changed 

 to z bv the relations, 



A ft 



x = e?, y = ze nt >, 



the resulting equation will be one in which the independent variable does not 

 appear and its order can be lowered by unity by 8.306. 



If y, -f-. r?i .... are assumed all to be of the same dimensions, and the 

 dx dx 



equation is homogeneous, the substitution: 



y _ e fudx 



will result in an equation in u and x of an order less by unity than the given 

 equation. ' 



8.310 Exact differential equations. A linear differential equation: 



where P, P Q , PI, ..... P n are functions of x is exact if: 

 dPi 



