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Question: Answered & Verified by Expert
If $x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)}$, then $|f(x y)|$ is equal to
MathematicsDifferential EquationsWBJEEWBJEE 2021
Options:
  • A $k e^{x^{2} / 2}$
  • B $\mathrm{ke}^{\mathrm{y}^{2} / 2}$
  • C $\mathrm{ke}^{\mathrm{x}^{2}}$
  • D $\mathrm{ke}^{\mathrm{y}^{2}}$
Solution:
2465 Upvotes Verified Answer
The correct answer is: $k e^{x^{2} / 2}$
$x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{x d y+y d x}{d x}=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y) d(x y)}{f(x y)}=x d x \Rightarrow \ln f(x y)=\frac{x^{2}}{2}+C \Rightarrow|f(x y)|=k e^{x^{2} / 2}$

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