Question

The solution of the differential equation $\frac{dy}{dx}-\frac{y+3x}{{\log }_e(y+3x)}+3=0$ is
(where $C$ is a constant of integration.)

• A. $x-{\log }_e(y+3x)=C$
• B. $x-\frac{1}{2}\left({\log }_e\right(y+3x){)}^2=C$
• C. $x-2{\log }_e(y+3x)=C$
• D. $y+3x-\frac{1}{2}({\log }_{e}x{)}^2=C$

Answer 1

The correct option is B $x-\frac{1}{2}\left({\log }_e\right(y+3x){)}^2=C$

Given: $\frac{dy}{dx}-\frac{y+3x}{\ln (y+3x)}+3=0$
$\Rightarrow \left(\frac{dy}{dx}\right)+3=\frac{y+3x}{\ln (y+3x)}$
$\Rightarrow \frac{1}{y+3x}(\frac{dy}{dx}+3)=\frac{1}{\ln (y+3x)}$

Put $\ln (y+3x)=t$
$\Rightarrow \frac{1}{y+3x}(\frac{dy}{dx}+3)=\frac{dt}{dx}$

$\therefore \frac{dt}{dx}=\frac{1}{t}$
$\Rightarrow dx-tdt=0$
$\Rightarrow x-\frac{t^2}{2}=C$
$\therefore x-\frac{1}{2}\left(\ln \right(y+3x){)}^2=C$