---
title: "Solve (x+y+1)dy/dx=1"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-x-y-1-dy-dx"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-x-y-1-dy-dx
site: BCT Engineering
---

# Solve (x+y+1)dy/dx=1

Mathematics I · Chapter 4: Ordinary Differential Equations and their Applications

- Asked in: 2068 Shrawan
- Video: [`(x+y+1) (dy)/(dx) =1`](https://www.youtube.com/watch?v=YRNnx80MUwk) — Doubtnut
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-x-y-1-dy-dx
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation (x + y + 1) dy/dx = 1.

## Short answer

Treat x as the dependent variable. The equation becomes dx/dy - x = y+1, a linear equation with integrating factor e^{-y}. Solving yields x + y + 2 = c e^y.

## Answer

(x + y + 1) dy/dx = 1

#### Take x as the dependent variable
dx/dy = x + y + 1 → dx/dy − x = y + 1

This is linear in x with P = −1, Q = y + 1.

#### Integrating factor
IF = e^(∫(−1)dy) = e^(−y)

#### Solution
x e^(−y) = ∫ (y + 1)e^(−y) dy

By parts: ∫ (y + 1)e^(−y) dy = −(y + 1)e^(−y) + ∫ e^(−y) dy = −(y + 1)e^(−y) − e^(−y) = −(y + 2)e^(−y)

x e^(−y) = −(y + 2)e^(−y) + c

x = −y − 2 + c eʸ

**x + y + 2 = c eʸ**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. (x + y + 1)dy/dx = 1 is linear in:
   A. both
   B. x with y independent
   C. y with x independent
   D. neither
   Answer: B. It gives dx/dy = x + y + 1. (Key point: Choice)

2. The standard form is:
   A. dx/dy - x = y + 1
   B. dy/dx - x = y + 1
   C. dx/dy + x = y + 1
   D. dx/dy - y = x + 1
   Answer: A. P = -1 and Q = y + 1. (Key point: Linear form)

3. The integrating factor is:
   A. e^y
   B. e^(-y)
   C. y + 1
   D. e^(-x)
   Answer: B. IF = e^(integral -1 dy). (Key point: Integrating factor)

4. The solution is:
   A. x + y + 2 = c e^y
   B. x - y - 2 = c e^y
   C. x + y + 1 = c e^y
   D. x + y + 2 = c e^(-y)
   Answer: A. x e^(-y) = -(y+2)e^(-y) + c. (Key point: Final solution)

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