Q110 / 180 · KCET 2022 · Mathematics

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

  • Correct answer
    A
    $\tan ^{-1}(x+y)=x+C$
  • B
    $\tan ^{-1}(x+y)=0$
  • C
    $\cot ^{-1}(x+y)=C$
  • D
    $\cot ^{-1}(x+y)=x+C$

Solution

$\frac{d y}{d x}=(x+y)^2$ Let $x+y=t \Rightarrow 1+\frac{d y}{d x}=\frac{d t}{d x}$ Eq. (i) can be written as $$ \begin{gathered} \frac{d t}{d x}-1=t^2 \\ \Rightarrow \quad \frac{d t}{d x}=t^2+1 \Rightarrow \frac{d t}{t^2+1}=d x \end{gathered} $$ $\begin{aligned} & \Rightarrow \int \frac{d t}{t^2+1}=\int d x \Rightarrow \tan ^{-1} t=x+C \\ & \Rightarrow \quad \tan ^{-1}(x+y)=x+C\end{aligned}$

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

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