EXAMPLES XX 845 



Then pju t - p 1 v l + p l v l log, ^ - 



EXAMPLES XX 



Solve the following equations, evaluating the constants by 

 using the special condition given in each case. 



(2) (arty + y) -j- = a; 2 !/ 2 + z 2 , given y = 0, when x =0 



^5" Sri' Sivenz/=2, when =0 



dw T/ 2 4- 1 



( 4 ) IT = ~* T g iven y = > when a; = 2 



CuC iC ~~ JL 



(5) -j- = sin (a; + y) - sin (a; - y), given t/ = ^, when x =0 



(6) J = sin 2 (x+y)- sin 2 (x-y), given r/ = ^, when x = 



(7) Cos 2 x -j- + y = 1, given t/ =0, when x =0 



(8) -^ + 2xy = #, given t/ = -, when x = 1 



n / ^ 



(9) (x*- y z ) /= xy, given t/ = 1, when # = 1 



cue 



(10) ^L = y\ x ~ y\ given w=l, when a: = 1 

 y dx x(x+ y) 



(11) a; 2 -^ = a; 2 + t/*, given w = -, when a? = 1 



</ ' * 



dw du 



(In Questions 9, 10, and 11 put y = IOT, then -^ = v + x ^-. 



Express the equation in terms of v and x, and solve 

 by separating the variables.) 



dv v TC 



(12) -p + - = sin x, given y = 0, when a; = - 

 cu7 a? ^ 



(13) ^ + t/ = sin 2a;, given y = 1, when a? - 



