Solve the following differential equation: `(dy)/(dx)+1=e^(x+y)`

Explanation

Given differential equaiton is `(dy)/(dx)+1=e^(x+y)`
On substructing x+y=t, we get `+(dy)/(dx)=(dt)/(dx)`
Eq. (i) becomes. `(dt)/(dx)=e^(t)`
`Rightarrow e^(-t)dt=dx`
`Rightarrow -e^(-t)=x+C`
`Rightarrow (-1)/(e^(x+y))=x+C`
`Rightarrow -1=(x+C)e^(x+y)`
`Rightarrow (x+C)e^(x+y)+1=0`