Q239 / 270 · NDA 2012 (Phase 2) · Mathematics

What is the general solution of the differential equation $x^2 d y+y^2 d x=0$ ?

  • A
    $x+y=c$
  • B
    $x y=c$
  • Correct answer
    C
    $c(x+y)=x y$
  • D
    None of the above where $c$ is the constant of integration

Solution

$\begin{aligned} & x^2 d y+y^2 d x=0 \\ & x^2 d y=-y^2 d x\end{aligned}$ $\frac{d x}{x^2}+\frac{d y}{y^2}=0$ on integrating, we get $-\frac{1}{x}-\frac{1}{y}=\mathrm{C}_1$ where $\mathrm{C}_1$ is constant of integration $\begin{aligned} \frac{x+y}{x y} & =-\mathrm{C}_1 \\ x+y & =-\mathrm{C}_1(x y)\end{aligned}$ $\Rightarrow \frac{1}{-\mathrm{C}_1}(x+y)=x y$ $\Rightarrow \quad \mathrm{C}(x+y)=x y \quad$ where $\mathrm{C}=\frac{1}{-\mathrm{C}_1}$

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

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