Q43 / 155 · WBJEE 2021 · Mathematics

If $x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)}$, then $|f(x y)|$ is equal to

  • Correct answer
    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

$x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{x d y+y d x}{d x}=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y) d(x y)}{f(x y)}=x d x \Rightarrow \ln f(x y)=\frac{x^{2}}{2}+C \Rightarrow|f(x y)|=k e^{x^{2} / 2}$

Hence, the correct answer is option: 'A'.

Relevant Questions