(X+y) dx + xdy =0

Answer 1

(x+y)dx = -x dy

-x dy /(x+y) dx = 1

-x / [ x(1 +y/x)] dx = 1

- dy /(1 +y/x) = dx

Let u =y/x then xu = y so x du/dx + u = dy/dx.

- (x du/dx + u) / ( 1+ u ) = 1

x du/dx + u = -(1 + u)

x du/dx + u = -1 - u

x du/dx + 2u = -1

(x/dx) du + 2u = -1

x/dx + (2u)/du = -1/du

x/dx = (-1 - 2u)/du

du/(-1-2u) = dx/x

(-1/2)log|-1 - 2u| = log|x| + C

Multiplying by -2:

log|-1-2u| = -2log|x| + C

log|-1-2u| = log|1/x^2| + C