Q249 / 270 · NDA 2018 (Phase 1) · Mathematics

What is the solution of $(1+2 x) d y-(1-2 y) d x=0$ ?

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

Solution

$\begin{aligned} & (1+2 x) d y-(1-2 y) d x=0 \Rightarrow(1+2 x) d y=(1-2 y) d x \\ & \Rightarrow \frac{d y}{1-2 y}=\frac{d x}{1+2 x} \end{aligned}$ Integrating both sides, $\int \frac{d y}{1-2 y}=\int \frac{d x}{1+2 x}$ $\begin{aligned} & \Rightarrow \frac{-1}{2} \log (1-2 \mathrm{y})=\frac{1}{2} \log (1+2 \mathrm{x})+\frac{1}{2} \log \mathrm{c} \\ & \Rightarrow \log (1-2 \mathrm{y})+\log (1+2 \mathrm{x})=\log \mathrm{c} \\ & \Rightarrow(1+2 \mathrm{x})(1-2 \mathrm{y})=\mathrm{c} \\ & \Rightarrow 1+2 \mathrm{x}-2 \mathrm{y}-4 \mathrm{xy}=\mathrm{c} \\ & \Rightarrow 2 \mathrm{x}-2 \mathrm{y}-4 \mathrm{xy}=\mathrm{c} \\ & \Rightarrow \mathrm{x}-\mathrm{y}-2 \mathrm{xy}=\mathrm{c} \end{aligned}$

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

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