THE INTEGRATING FACTOR 823 



dz .dw 



and -T- - ef and -r- = P 



dw dx 



dz dz dw 

 but 



<li dw dx 



Then 



Thus, if the differential equation is multiplied throughout 

 by e^* 6 , the left-hand side becomes the result which would 

 be obtained by differentiating ye ** , 



and 



or 



x 



** 



Integrating, yr* ** = I Qe> dx + Const 



e* x is known as the integrating factor. 



flll 



Example 1. Solve the equation - + 2#y = a;. 



Now IP dx = 2 



The integrating factor is f* 



Hence ye** = \xe**dx + c 



To find ^xef 2 dx, put # 2 = z 



Then - dz = 2x dx 



If 1 



and the integral becomes -\e* dz = -t 



Therefore ye? = -^ + c 



2 



and w = - -f ce~ z 



* 2 



