The solution of the differential equation

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.)

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

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

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

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


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)

$\frac{\mathrm{dz}}{\mathrm{dx}}=\frac{1}{\mathrm{z}}$

$\Rightarrow \mathrm{z} \mathrm{dz}=\mathrm{dx} \Rightarrow \frac{\mathrm{z}^{2}}{2}=\mathrm{x}+\mathrm{C}$

$\Rightarrow \frac{1}{2} \ell \mathrm{n}^{2}(\mathrm{y}+3 \mathrm{x})=\mathrm{x}+\mathrm{C}$

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

 

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