322 PRACTICAL MATHEMATICS 



Example 2. Solve the equation x*y z - 4>x 2 = (x 2 y 2 - 9*/ 2 ) ^ 



Now a% 2 - 4) = y*(x* - 9) ^ 



# 2 ?/ 2 ^ 



-2 



Example 3. Solve the equation 



/77/ 



sin 2 x sin 2 y cos 2 x cos 2 y -j- = 

 Now sin 2 x sin 2 ?/ = cos 2 x cos 2 i/ -r- 



and tan 2 x = cot 2 ?/ -p 



Itan 2 x dx = Icot 2 y dy + c 



j (sec 2 a? - 1) da; = I (cosec 2 ?/ - 1) d?/ + c 



tan x x = cot y y + c 

 tan x + cot y x + y = c 



166. The Use of the Integrating Factor. The integrating factor 

 is used to solve differential equations of the form, or differential 

 equations which can be reduced to the form, 



where P and Q can either be constants or functions of x, but they 

 must be independent of y. 



Now e*** 1 when differentiated with respect to x gives P^ Pd 

 For z = e w where w = IP dx 



