Question

Solution of the differential equation
$e^x\left(\sin y\right)dx+(1+e^x)\cos ydy=0$ is

• A. $(e^x+1)\sin y=k$
• B. $(e^x-1)\ln \sin y=k$
• C. $(e^x+1)\sec y=k$
• D. $(e^x+1)\cos y=k$

Answer 1

The correct option is A $(e^x+1)\sin y=k$

$e^x\left(\sin y\right)dx+\left(1+e^x\right)dy=0$

$\Rightarrow e^x\sin ydy=-\left(1+e^x\right)\cos ydy$

$\Rightarrow \frac{e^x}{1+e^x}dx=-\frac{\cos y}{\sin y}dy$

$\Rightarrow {\int }^{}\frac{e^x}{1+e^x}dx={\int }^{}-\frac{\cos y}{\sin y}dy={\int }^{}\cot ydy$

$\Rightarrow {\int }^{}\frac{dt}{t}=-\log \left|\mathrm{siny}\right|+\log k$

$\Rightarrow \log \left|t\right|=-\log \left|\mathrm{siny}\right|+\log k$

$\Rightarrow \log \left(1+e^x\right)=-\log \mathrm{siny}+\log k$

$\Rightarrow \log \left(1+e^x\right)=\log \frac{k}{\mathrm{siny}}$

$\Rightarrow 1+e^x=\frac{k}{\mathrm{siny}}$

$\Rightarrow \left(1+e^x\right)\mathrm{siny}=k$