Question

The general solution of the differential equation $(x+y+1)dy=dx$ is (where $C$ is a constant of integration)

• A. $|x+y+2|=C{e}^y$
• B. $x+y+4=C\ln |y|$
• C. $\ln |x+y-2|=Cy$
• D. $\ln |x+y+2|=C-|y|$

Answer 1

The correct option is A $|x+y+2|=C{e}^y$

Putting $x+y+1=u,$ we have $du=dx+dy$ and the given equations reduces to
$u\left(du-dx\right)=dx$
$\Rightarrow \frac{udu}{u+1}=dx$
On integrating both sides
$\Rightarrow u-\ln |u+1|=x+c$
$\Rightarrow \ln |x+y+2|=y+1-c;$ where $(1-c=c_1)$
$\Rightarrow |x+y+2|=e^{y+c_1}$
$\Rightarrow |x+y+2|=C{e}^y;$
$\because (e^{c_1}=C)$