Question

Solve: $\frac{dy}{dx}+1=e^{x+y}$.

Answer 1

Let $x+y=t$

$\Rightarrow 1+\frac{dy}{dx}=\frac{dt}{dx}$

Substituting in question, we get ;

$\frac{dt}{dx}=e^t$

$\Rightarrow e^{-t}dt=dx$

Integrate on both sides

$\Rightarrow \frac{e^{-t}}{-1}=x+c$

$\Rightarrow x+e^{-t}+c=0$

$\Rightarrow x+e^{-(x+y)}+c=0$ ('c' is a constant)