Question

Solve the differential equation `"dy"/"dx" + 1` = e^{x + y}.

Answer 1

Given that: `"dy"/"dx" + 1` = e^{x + y} 

Put x + y = t

∴ `1 + "dy"/"dx" = "dt"/"dx"`

∴ `"dt"/"dx"` = e^t

⇒ `"dt"/"e"^"t"` = dx

⇒ `"e"^-"t" "dt"` = dx

Integrating both sides, we have 

`int"e"^-1 "dt" = int "d"x`

⇒ `-"e"^"t"` = x + c

⇒ `-"e"^(-(x + y)` = x + c

⇒`(-1)/"e"^(x + y)` = x + c

⇒ (x + c)e^{x + y} = –1

Hence, the required solution is (x + c).e^{x + y} + 1 = 0.