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: , 3
Solution:
$\ell \mathrm{n}(\mathrm{y}+3 \mathrm{x})=\mathrm{z}$ (let)
$\frac{1}{y+3 x} \cdot\left(\frac{d y}{d x}+3\right)=\frac{d z}{d x}$ ......(1)
$\frac{d y}{d x}+3=\frac{y+3 x}{\ln (y+3 x)}$ (given)
$\frac{\mathrm{dz}}{\mathrm{dx}}=\frac{1}{\mathrm{z}}$
$\Rightarrow \mathrm{z} \mathrm{dz}=\mathrm{dx} \Rightarrow \frac{\mathrm{z}^{2}}{2}=\mathrm{x}+\mathrm{C}$
$\Rightarrow \frac{1}{2} \ell \mathrm{n}^{2}(\mathrm{y}+3 \mathrm{x})=\mathrm{x}+\mathrm{C}$
$\Rightarrow \mathrm{x}-\frac{1}{2}(\ell \mathrm{n}(\mathrm{y}+3 \mathrm{x}))^{2}=\mathrm{C}$