DIFFERENTIAL EQUATIONS 163 



8.011 Equations of the form: 



dy _ a'x + b'y + c' 

 dx ax -f by + c 



If ab' a'b ^ o, the substitution 



x = x' + p, y = y' + q, 

 where 



ap + bq + c = o, 



a'p + b'q + c' = o, 



renders the equation homogeneous, and it may be solved by 8.010. 



If ab' - a'b = o and b' ^ o, the change of variables to either x and 2 or y 

 and z by means of 



z = ax + by, 



will make the variables separable (8.001). 



8.020 Exact differential equations. The equation, 



P(x, y)dx + Q(x, y)dy = o, 

 is exact ir, 



dx dy 

 The solution is: 



(*, y)dx + / { Q(x, y) - |- I P(x, y}dx \ dy = C, 



or 



r 



(x, y)dy + I { P(x, y) - - / Q(x, y)dy \ dx = C. 



8.030 Integrating factors. v(x, y) is an integrating factor of 



P(x, y) dx + Q(x, y) dy = o, 

 if 



8.031 If one only of the functions Px + Qy and Px Qy is equal to o, the 

 reciprocal of the other is an integrating factor of the differential equation. 



8.032 Homogeneous equations. If neither Px + Qy nor Px - Qy is equal to o, 



is an integrating factor of the equation if it is homogeneous. 



