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)

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