Question

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

Answer 1

$\frac{dy}{dx}+1=e^{x+y}$ .....(1)
Let x + y = t
$\Rightarrow 1+\frac{dy}{dx}=\frac{dt}{dx}$
Substituting the value of x + y = t and $1+\frac{dy}{dx}=\frac{dt}{dx}$ in (1), we get
$$\begin{gathered} \frac{dt}{dx}=e^t \\ \Rightarrow e^{-t}dt=dx \\ \Rightarrow -e^{-t}=x+C \\ \Rightarrow -e^{-\left(x+y\right)}=x+C\left[\because t=x+y\right] \end{gathered}$$