The solution of the differential equation


The solution of the differential equation $\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0$ is :

(where $C$ is a constant of integration.)

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

  2. (2) $x-\log _{e}(y+3 x)=C$

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

  4. (4) $x-2 \log _{e}(y+3 x)=C$

Correct Option: 1


Let $y+3 x=t$

$\Rightarrow \frac{d y}{d x}+3=\frac{d t}{d x}$

Putting these value in given differential equation

$\frac{d t}{d x}=\frac{t}{\log _{e} t}$

$\Rightarrow \int \frac{\log _{e} t}{t} d t=\int d x$

$\Rightarrow \frac{\left(\log _{e} t\right)^{2}}{2}=x-C$

$\Rightarrow x-\frac{1}{2}(\ln (y+3 x))^{2}=C$

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