Question

Solve the differential equation ${(x+y)}^2\frac{dy}{dx}=a^2$

Answer 1

Put $x+y=t\Rightarrow 1+\frac{dy}{dx}=\frac{dt}{dx}$

$\Rightarrow \frac{dt}{dx}=\frac{a^2}{t^2}+1=\frac{a^2+t^2}{t^2}$

$\Rightarrow \int \frac{t^2}{a^2+t^2}dt=\int dx$

$\Rightarrow t-a{\tan }^{-1}\frac{t}{a}=x+c$, where $c$ is any constant.

$\Rightarrow y-a{\tan }^{-1}\frac{x+y}{a}=c$