Q94 / 180 · COMEDK 2014 · Mathematics

The solution of $\left(2 x-10 y^{3}\right) \frac{d y}{d x}+y=0$ is

  • Correct answer
    A
    $x y^{2}=2 y^{5}+C$
  • B
    $y x^{2}=2 y^{5}+C$
  • C
    $x^{2} y^{2}=2 y^{5}+C$
  • D
    None of these

Solution

Given, differential equation is $$ \begin{aligned} &\left(2 x-10 y^{3}\right) \frac{d y}{d x}+y=0 \Rightarrow \frac{d y}{d x}=\frac{-y}{2 x-10 y^{3}} \\ \Rightarrow \quad & \frac{d x}{d y}=\frac{2 x-10 y^{3}}{-y}=\frac{-2 x}{y}+10 y^{2} \\ \Rightarrow \quad & \frac{d x}{d y}+\frac{2}{y} x=10 y^{2} \end{aligned} $$ Compare with linear differential equation $$ \frac{d x}{d y}+P x=Q \Rightarrow \mathrm{IF}=e^{\int \frac{2}{y} d y}=e^{\log y^{2}}=y^{2} $$ $\therefore$ Required solution is $x \cdot y^{2}=\int 10 y^{2} \cdot y^{2} d y+C$ i.e. $$ x y^{2}=2 y^{5}+C $$

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

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