Question

The general solution of the differential equation $(e^x+1)ydy=(y+1)e^{x}dx$ is (where $c$ is a constant of integration)

• A. $(y-1)|e^x-1|=c{e}^y$
• B. $|y-1|(e^x+1)=2c{e}^y$
• C. $(y+1)(e^x-1)=4c{e}^{|y|}$
• D. $|y+1|(e^x+1)=e^y|c|$

Answer 1

The correct option is D $|y+1|(e^x+1)=e^y|c|$

The given differential equation is $(e^x+1)ydy=(y+1)e^{x}dx$

$\frac{ydy}{(y+1)}=\frac{e^{x}dx}{(e^x+1)}$

Integrating both sides, we get

$\Rightarrow y-\ln |y+1|=\ln |e^x+1|+\ln |k|\Rightarrow y=\ln |(y+1)(e^x+1)k|$

$\Rightarrow |y+1|(e^x+1)=e^y|c|;$ where $c=\frac{1}{k}$