Q120 / 160 · TS EAMCET 2024 (09 May Shift 2) · Mathematics

The general solution of the differential equation $\frac{d y}{d x}=\frac{2 y^2+1}{2 y^3-4 x y+y}$ is

  • Correct answer
    A
    $4 x y^2+2 x=y^4+y^2+c$
  • B
    $2 x y^2+x=y^4-y^2+c$
  • C
    $4 x y^2-2 x=y^4+y^2+c$
  • D
    $4 x y^2+2 x=y^4-y^2+c$

Solution

$\frac{d x}{d y}=\frac{2 y^3+y-4 x y}{2 y^2+1}=y-\frac{4 x y}{2 y^2+1}$ $\begin{aligned} & \frac{d x}{d y}+\frac{4 x y}{2 y^2+1}=y \\ & I F=e^{\int \frac{-4 y}{2 y^2+1}}=e^{\ln \left(2 y^2+1\right)}=2 y^2+1 \\ & x\left(2 y^2+1\right)=\int\left(2 y^3+y\right) d y=\frac{y^4}{2}+\frac{y^2}{2}+C \\ & \Rightarrow 4 x y^2+2 x=y^4+y^2+C\end{aligned}$

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

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