Question

The general solution of $\frac{dy}{dx}={(x+y)}^2,$ where $c$ is the constant of integration, is

• A. $x+y=\tan (x+c)$
• B. $x+c=\tan (x+y)$
• C. $y=\tan x+c$
• D. ${(x+y)}^2=\tan (x+c)$

Answer 1

The correct option is A $x+y=\tan (x+c)$

$\frac{dy}{dx}={(x+y)}^2\cdots \left(i\right)$
putting $x+y=v,$ we have
$1+\frac{dy}{dx}=\frac{dv}{dx}\Rightarrow \frac{dv}{dx}=v^2+1$
$\Rightarrow \int \frac{dv}{v^2+1}=\int dx$
$\Rightarrow {\tan }^{-1}v=x+c$
$\Rightarrow x+y=\tan (x+c)$