Q245 / 270 · NDA 2023 (Phase 1) · Mathematics

A solution of the differential equation $\left(\frac{d y}{d x}\right)^2-x \frac{d y}{d x}=0$ is

  • A
    $y=2 x$
  • B
    $y=2 x+4$
  • C
    $y=x^2-1$
  • Correct answer
    D
    $y=\frac{\left(x^2-2\right)}{2}$

Solution

$\left(\frac{d y}{d x}\right)^2-x\left(\frac{d y}{d x}\right)=0$ $\frac{d y}{d x}\left[\frac{d y}{d x}-\right]=$ $\frac{d y}{d x}=0$ or $\frac{d y}{d x}=x$ New $d y=x d x$ $\Rightarrow \int d y=\int x d x$ $y=\frac{x^2}{2}+C \Rightarrow y=\frac{x^2+2 C}{2}$ $y=\frac{x^2}{2}+C \Rightarrow y=\frac{x^2+2 C}{2}$ Comparing with options, $y=\frac{x^2-2}{2}$

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

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