---
title: "Solve (1+y²)dx = (tan⁻¹y − x)dy"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-1-plus-y2-dx-atan-y-minus-x"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-1-plus-y2-dx-atan-y-minus-x
site: BCT Engineering
---

# Solve (1+y²)dx = (tan⁻¹y − x)dy

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

- Asked in: 2074 Asoj; 2070 Asar (asked 2 times)
- Video: [Solve: (1+y^2)dx=(tan^-1y-x)dy](https://www.youtube.com/watch?v=NitVyEw4Rrs) — RK Eduworld (Classic)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/de-1-plus-y2-dx-atan-y-minus-x
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve: (1+y²)dx = (tan⁻¹y − x)dy.

## Short answer

Rewrite as linear in x: dx/dy + x/(1+y²) = tan⁻¹y/(1+y²). Integrating factor e^(tan⁻¹y). Solution: x = tan⁻¹y − 1 + C e^(−tan⁻¹y).

## Answer

The equation is linear in x when y is the independent variable.

#### Rewrite as a linear equation in x
dx/dy = (tan⁻¹y − x)/(1 + y²) → dx/dy + x/(1 + y²) = tan⁻¹y/(1 + y²)

This is linear in x: dx/dy + Px = Q with P = 1/(1 + y²), Q = tan⁻¹y/(1 + y²).

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

#### Solution
x e^(tan⁻¹y) = ∫ [tan⁻¹y/(1 + y²)] e^(tan⁻¹y) dy

Put t = tan⁻¹y, dt = dy/(1 + y²):

∫ t e^t dt = t e^t − e^t + C = (t − 1)e^t + C

So x e^(tan⁻¹y) = (tan⁻¹y − 1)e^(tan⁻¹y) + C

**x = tan⁻¹y − 1 + C e^(−tan⁻¹y)**

## Check yourself

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

1. Why is (1+y^2)dx = (tan^-1 y - x)dy treated as linear?
   A. It is homogeneous
   B. It is exact
   C. It is linear in y with x independent
   D. It is linear in x with y independent
   Answer: D. Rewriting as dx/dy + x/(1+y^2) = tan^-1 y/(1+y^2) is linear in x. (Key point: Recognising linearity)

2. What is the integrating factor?
   A. tan^-1 y
   B. e^(tan^-1 y)
   C. 1 + y^2
   D. e^y
   Answer: B. IF = e^(integral dy/(1+y^2)) = e^(tan^-1 y). (Key point: Integrating factor)

3. Which substitution simplifies the right-hand integral?
   A. t = 1 + y^2
   B. t = e^y
   C. t = y^2
   D. t = tan^-1 y, giving integral of t e^t dt
   Answer: D. With t = tan^-1 y, dt = dy/(1+y^2) and the integral becomes t e^t dt. (Key point: Substitution)

4. The general solution is:
   A. x = e^(-tan^-1 y)(tan^-1 y + 1)
   B. x = tan^-1 y - 1 + C e^(-tan^-1 y)
   C. x = (tan^-1 y - 1) + C
   D. x = tan^-1 y + 1 + C e^(tan^-1 y)
   Answer: B. Integration by parts gives (t - 1)e^t + C; dividing by the IF yields the stated result. (Key point: Final solution)

## Related in Mathematics I

- [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx − 2y tan x = y² tan x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-bernoulli-2y-tan-x) — Mathematics I, chapter 4, asked 6×
- [Solve (1+x²)dy/dx+y=e^(tan⁻¹x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-x2-dy-dx-etan) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx + y tan x = sec x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-linear-y-tan-x-sec-x) — Mathematics I, chapter 4
- [Solve x dy/dx + y log y = xy eˣ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-dy-dx-y-log-y) — Mathematics I, chapter 4, asked 4×

## Related in other subjects

- [RK-4: dy/dx = (1+x²)y](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Ark4-y-1-x2) — Numerical Method, chapter 6
- [Polar coordinates: ∫∫(x²+y²) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-x2-y2-dy) — Mathematics, chapter 2, asked 5×
- [Evaluate ∫∫ e^(y/x) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-y-x-dy) — Mathematics, chapter 2
- [Evaluate ∫∫ e^(x²) dx dy](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-x2-dx-dy) — Mathematics, chapter 2