Q69 / 150 · MHT CET 2008 · Mathematics

The solution of the differential equation $\frac{d y}{d x}=\frac{x-y+3}{2(x-y)+5}$ is

  • A
    $2(x-y)+\log (x-y)=x+c$
  • B
    $2(x-y)-\log (x-y+2)=x+c$
  • Correct answer
    C
    $2(x-y)+\log (x-y+2)=x+c$
  • D
    None of the above

Solution

Given differential equation is $$ \frac{d y}{d x}=\frac{x-y+3}{2(x-y)+5} $$ Let $\quad x-y=v \Rightarrow \frac{d y}{d x}=1-\frac{d v}{d x}$ $\therefore$ $$ \frac{d v}{d x}=\frac{v+2}{2 v+5} $$ $$ \begin{array}{l} \Rightarrow \quad \int \frac{2 v+5}{v+2} d v=\int d x \\ \Rightarrow \quad \int\left(2+\frac{1}{v+2}\right) d v=\int d x \end{array} $$ $\Rightarrow \quad 2 v+\log (v+2)=x+c$ $\Rightarrow 2(x-y)+\log (x-y+2)=x+c$

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

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