Question

Solve (dy)/(dx)=(x-y+2)/(x-y-2)using an appropriate substitution.

Answer 1

Solution

New variable u = x-y  =>   du/dx = 1-dy/dx  =>  du/dx = 1-(u+2)/(u-2)  => du/dx = -4/(u-2)

It is separable differential equation

(u-2)du = -4dx   =>  ∫(u-2)du = -4∫dx   =>  u²/2-2u = -4x+C  =>  u²-4u = -8x+C  =>  

$u(x)=2\pm\sqrt{4-8x+C}$

$y(x)=x-2\pm2\sqrt{1-2x+C}$