Question

Solve $(x+y+1)\frac{dy}{dx}=1$.

Answer 1

Re-arrangement of the terms give

$x+y+1=\frac{dx}{dy}$

$\frac{dx}{dy}-x=y+1$

$IF=e^{\int 1.dy}$

$=e^y$

Hence

$e^y\frac{dx}{dy}-e^{y}x=e^{y}y+e^y$

$e^{y}dx+e^y.xdy=e^{y}y.dy+e^{y}dy$

$\int d(e^y.x)=\int e^{y}ydy+\int e^{y}dy$

$e^y.x=y{e}^y-\int e^y.dy+\int e^{y}dy$

$e^{y}x=y{e}^y+C$

$e^y(x-y)=C$