The solution of the differential equation `(dy)/(dx)+1=e^(x+y)`, is

Options

A. `(x+y)e^(x+y)=0`
B. `(x+C)e^(x+y)=0`
C. `(x-C)e^(x+y)=1`
D. `(x+C)e^(x+y)+1=0`

Answer

D

Explanation

We have,
`(dy)/(dx)+1=e^(x+y)`
Let `x+y=v`. Then, `1+(dy)/(dx)=(dv)/(dx)`
`therefore" "(dy)/(dx)+1=e^(x+y)`
`rArr" "(dv)/(dx)=e^(v)`
`rArr" "e^(-v)dv=dx`
`rArr" "inte^(-v)dv=intdx`
`rArr" "-e^(-v)=x+C`
`rArr" "-e^(-(x+y))=x+C rArr (x+C) e^(x+y)+1=0`