Question

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

• A. $\frac{1}{x+y} = c$
• B. $\sin^{-1} (x + y) =x +c$
• C. $\tan^{-1} (x +y) = c$
• D. $\tan^{-1} (x +y) = x +c$

Answer 1

Answer: (D)

$\tan^{-1} (x +y) = x +c$

Answer 2

Let $x + y = t \: \Rightarrow \: 1 + \frac{dy}{dx} = \frac{dt}{dx}$ $\frac{dt}{dx}-1=t^{2} \Rightarrow \frac{dt}{dx} =t^{2} +1 \Rightarrow \int\frac{dt}{t^{2}+1} = \int dx$ $\Rightarrow \tan^{-1}t =x+c \Rightarrow \tan^{-1}\left(x+y\right)=x+c$