Q230 / 270 · NDA 2020 · Mathematics

The function $u(x, y)=c$ which satisfies the differential equation $x(d x-d y)+y(d y-d x)=0$, is

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

Solution

$x(d x-d y)+y(d y-d x)=0$ On integrating both sides, we have $\begin{aligned} & \int x(d x-d y)+\int y(d y-d x)=C(\text { where } C=\text { constant }) \\ & \frac{x^2}{2}-x y+\frac{y^2}{2}-x y=C \\ & x^2+y^2-2 x y=C \\ & \text { Or, } x^2+y^2=2 x y+C \end{aligned}$

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

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