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Question: Answered & Verified by Expert
If $x \frac{d y}{d x}+y=x \frac{f(x y)}{f^{\prime}(x y)}$, then $|f(x y)|$ is equal to
MathematicsDifferential EquationsWBJEEWBJEE 2022
Options:
  • A $\mathrm{Ce}^{\mathrm{x}^2 / 2}$
  • B $\mathrm{Ce}^{\mathrm{x}^2}$
  • C $\mathrm{Ce}^{2 x^2}$
  • D $\mathrm{Ce}^{\mathrm{x}^2 / 3}$
Solution:
2682 Upvotes Verified Answer
The correct answer is: $\mathrm{Ce}^{\mathrm{x}^2 / 2}$
$x \frac{d y}{d x}+y=x \frac{f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{d(x y)}{d x}=x \frac{f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y)}{f(x y)} d(x y)=x d x \Rightarrow \ln |f(x y)|=\frac{x^2}{2}+k \Rightarrow f(x y)=C e^{\frac{x^2}{2}}$

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