Q118 / 160 · AP EAMCET 2006 · Mathematics

The solution of $\left(x^2+y^2\right) d x=2 x y d y$ is :

  • Correct answer
    A
    $c\left(x^2-y^2\right)=x$
  • B
    $c\left(x^2+y^2\right)=x$
  • C
    $c\left(x^2-y^2\right)=y$
  • D
    $c\left(x^2+y^2\right)=y$

Solution

$\because \quad \frac{x^2+y^2}{2 x y}=\frac{d y}{d x}$ Put $y=v x$ and $\frac{d y}{d x}=v+x \frac{d v}{d x}$ $\therefore \quad v+x \frac{d v}{d x}=\frac{x^2+v^2 x^2}{2 x^2 v}$ $\Rightarrow \quad v+x \frac{d v}{d x}=\frac{1+v^2}{2 v}$ $\Rightarrow \quad \frac{x d v}{d x}=\frac{1+v^2-2 v^2}{2 v}$ $\Rightarrow \quad \frac{2 v}{1-v^2} d v=\frac{1}{x} d x$ $\Rightarrow \quad-\log \left(1-v^2\right)=\log x+\log c$ $\Rightarrow \quad-\log \left(\frac{x^2-y^2}{x^2}\right)=\log x+\log c$ $\Rightarrow \quad-\log \left(x^2-y^2\right)+2 \log x=\log x+\log c$ $\Rightarrow \quad \log x=\log c+\log \left(x^2-y^2\right)$ $\Rightarrow \quad x=c\left(x^2-y^2\right)$

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

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