Question
Wed November 23, 2011 By: Aakarsh Sharma

solve the diff. equation dx/x+y=dy/x-y

Expert Reply
Wed November 23, 2011
dx/x+y = dy/x-y
 
or,  dy/dx = x-y/x+y ...............1
 
let y=tx, where t is another third variable like x and y.
 
differentiate both the sides w.r.t. x
 
dy/dx = t + x(dt/dx)......................2
 
putting eqn 2 in 1 we get
 
t+xdt/dx=x-tx/x+tx
 
xdt/dx = [1-t/1+t] -t
 
xdt/dx = 1-t -t - t2/1+t
 
(t+1)dt/(1-2t- t2) = dx/x
 
let u=1-2t-t2, So, du/dt=-2-2t=-2(1+t)
so (1+t)dt=-du/2
 
So, -du/2u = dx/x
 
So, -lnu=2lnx
 
so, x2=1/u
 
so, x2=1/1-2t-t2
now t=y/x replacing its value
 
x2= 1/1-2(y/x)-(y/x)2
 
x2= x2/x2-2xy-y2
 
or, x2-2xy-y2=1
this is the answer
 
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