172 



8.211 



where X is any function of x. 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



V(x) = CX k X, 



y = cxk F(e~^j x ' 



8.220 The differential equation: 

 (a + bx)" + ( a + bx)"- 1 ^ 



(a 



a n ., + a n y = V(x), 



may be reduced to the homogeneous linear equation (8.200) by the change of 



variable 



z = a + ox. 



It may be reduced to a linear equation with constant coefficients by the 



change of variable: 



e z = a + bx. 



8.230 The general linear equation. General form: 



where P Q , PI, ..... , P n , V are functions of x only. 

 The complete solution is the sum of: 



(a) The complementary function, which is the general solution of the equation 

 with V = o, and containing n arbitrary constants, and 



(b) The particular integral. 



8.231 Complementary Function. If y i} y 2 , . . . . , y n are n independent solu- 



tions of 8.230 with V = o, the complementary function is 

 y = ciy\ + c z yz + ...... + c n y n . 



The conditions that yi, y 2 , . . . . , y n be n independent solutions is that the 



determinant A ^ o. 

 A = 



When A = o: 



A = 



- 



