Q109 / 180 · KCET 2019 · Mathematics

The integrating factor of the differential equation \( \left(2 x+3 y^{2}\right) d y=y d x(y>0) \) is

  • A
    \( -\frac{1}{y^{2}} \)
  • Correct answer
    B
    \( \frac{1}{y^{2}} \)
  • C
    \( e^{\frac{1}{y}} \)
  • D
    \( \frac{1}{x} \)

Solution

\( (\mathrm{B}) \) \( \Rightarrow \frac{\left.2 x+3 y^{2}\right) \mathrm{dy}}{y}=\mathrm{y} \mathrm{dx} \) \( \Rightarrow \frac{2 x}{y}+3 y=\frac{d x}{d y} \) \( \Rightarrow \frac{d x}{d y}-\frac{2 x}{y}=3 y \) If \( =e^{\int-\frac{2}{y} d y}=e^{-2 \log y}=e^{\log y^{-2}}=\frac{1}{y^{2}} \)

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

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