Q117 / 160 · AP EAMCET 2008 · Mathematics

The solution of the differential equation $\frac{d y}{d x}=\frac{x y+y}{x y+x}$ is

  • A
    $x+y-\log \left(\frac{c y}{x}\right)$
  • B
    $x+y=\log (c x y)$
  • C
    $x-y-\log \left(\frac{c x}{y}\right)$
  • Correct answer
    D
    $y-x=\log \left(\frac{c x}{y}\right)$

Solution

Given differential equation is $$ \begin{aligned} \frac{d y}{d x} & =\frac{x y+y}{x y+x} \\ \Rightarrow \quad \frac{d y}{d x} & =\frac{y(1+x)}{x(1+y)} \\ \Rightarrow \quad \frac{(1+y)}{y} d y & =\frac{(1+x)}{x} d x \\ \Rightarrow \quad \int\left(\frac{1}{y}+1\right) d y & =\int\left(\frac{1}{x}+1\right) d x \\ \Rightarrow \quad \log y+y & =\log x+x+\log c \\ \Rightarrow \quad y-x & =\log \left(\frac{c x}{y}\right) \end{aligned} $$

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

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