Question

The solution of the differential equation ydx + (x + xy)dy = 0 is ______.

Answer 1

The solution of the differential equation ydx + (x + xy)dy = 0 is xy = ce^–y.

Explanation:

The given differential equation is ydx + (x + xy)dy = 0

⇒ (x + xy)dy = – ydx 

⇒ x(1 + y)dy = – ydx

⇒ `(1 + y)/y  "d"y = - 1/x  "d"x`

Integrating both sides, we get

`int (1 + y)/y  "d"y = - int 1/x "d"x`

⇒ `int(1/y + 1)"d"y = -int 1/x "d"x`

⇒ log y + y = – log x + log c

⇒ log x + log y + log e y = log c

⇒ log(xy . e^y) = log c

∴ xy = ce^–y