Q120 / 160 · AP EAMCET 2024 (20 May Shift 1) · Mathematics

The general solution of the differential equation $\left(y^2+x+1\right) d y=(y+1) d x$ is

  • A
    $x+2+(y+1) \log (y+1)^2=y+c$
  • B
    $x+2+\log (y+1)^2=\frac{y}{y+1}+c$
  • C
    $\frac{x}{y+1}=\log (y+1)^2+y+c$
  • Correct answer
    D
    $\frac{x+2}{y+1}+\log (y+1)^2=y+c$

Solution

$\frac{d x}{d y}=\frac{y^2+x+1}{y+1} \Rightarrow \frac{d x}{d y}-\frac{x}{y+1}=\frac{y^2+1}{y+1}$ If $=e^{\int-\frac{1}{y+1} d y}=\frac{1}{y+1} \frac{x}{y+1}=\int \frac{y^2+1}{(y+1)^2} d y=\int\left(1-\frac{2 y}{(y+1)^2}\right) d y$ $\frac{x}{y+1}=y-2 \log |y+1|-\frac{2}{y+1}+c$ $\frac{x+2}{y+1}+2 \log (y+1)=y+c$

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

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