Question:
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.)
Correct Option: 1
Solution:
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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