Question

The general solution of the differential equation $\frac{dy}{dx}=\frac{x-y+1}{x+y+1}$ is
(where $c$ is constant of integration)

• A. $xy+\frac{y^2}{2}-\frac{x^2}{2}+y-x=c$
• B. $xy+y-x=c$
• C. $xy+\frac{y^2}{2}-\frac{x^2}{2}-y+x=c$
• D. $xy-y-x=c$

Answer 1

The correct option is A $xy+\frac{y^2}{2}-\frac{x^2}{2}+y-x=c$

Given : $\frac{dy}{dx}=\frac{x-y+1}{x+y+1}$
$\Rightarrow xdy+ydy+dy=xdx-ydx+dx$
$\Rightarrow (xdy+ydx)+ydy-xdx+dy-dx=0$
$\Rightarrow (xdy+ydx)+\frac{2y}{2}dy-\frac{2x}{2}dx+dy-dx=0$
$\Rightarrow d\left(xy\right)+\frac{d\left(y^2\right)}{2}-\frac{d\left(x^2\right)}{2}+dy-dx=0$
Integrating both sides, we get
$xy+\frac{y^2}{2}-\frac{x^2}{2}+y-x=c$