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The solution of the differntial equation \frac{dy}{dx}-\frac{y +3x}{\log_{e} (y+3x)}+3=0 is: (where C is a constant of intergration)
Option: 1 x - \frac{1}{2}\left ( \log _{e}(y+3x) \right )^{2}=C
Option: 2 x - \log _{e}(y+3x)=C
Option: 3 y +3x-\frac{1}{2}(\log _{e}x^{2})=C
Option: 4 x-2 \log _{e} (y+3x)=C

Answers (1)

best_answer

Let

ln (y + 3x) = z 

\begin{aligned} &\frac{1}{y+3 x}\left(\frac{d y}{d x}+3\right)=\frac{d z}{d x}\\ &\frac{d y}{d x}+3=\frac{y+3 x}{\ell n(y+3 x)}(\text { 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} \end{aligned}

\\\Rightarrow \mathrm{z} \mathrm{dz}=\mathrm{dx} \Rightarrow \frac{\mathrm{z}^{2}}{2}=\mathrm{x}+\mathrm{C} \\ \Rightarrow \frac{1}{2} \ln ^{2}(\mathrm{y}+3 \mathrm{x})=\mathrm{x}+\mathrm{C} \\ \Rightarrow \mathrm{x}-\frac{1}{2}(\ln (\mathrm{y}+3 \mathrm{x}))^{2}=\mathrm{C}

Posted by

himanshu.meshram

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