---
title: "Solve x dy − y dx = √(x²+y²) dx"
subject: "Mathematics I"
chapter: 4
key: "semester_1/mathematics_i:de-x-dy-minus-y-dx-sqrt"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/de-x-dy-minus-y-dx-sqrt
site: BCT Engineering
---

# Solve x dy − y dx = √(x²+y²) dx

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

- Asked in: 2069 Asar
- Video: [Differential equation of x DY - y DX = root x square + y square DX](https://www.youtube.com/watch?v=6vFV9QQ8owU) — Ankit Maths Lab
- 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-dy-minus-y-dx-sqrt
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-4

## Question

Solve the differential equation x dy − y dx = √(x²+y²) dx.

## Short answer

Substituting y = vx reduces x dy − y dx = √(x²+y²) dx to dv/√(1+v²) = dx/x. Integration gives y + √(x²+y²) = cx², or equivalently 2cy = c²x² − 1.

## Answer

x dy − y dx = √(x² + y²) dx

The equation is homogeneous (take x > 0).

#### Substitution
Put y = vx, dy = v dx + x dv. Then x dy − y dx = x² dv, and the equation becomes

x² dv = x√(1 + v²) dx → dv/√(1 + v²) = dx/x

#### Integrate
sinh⁻¹ v = ln x + ln c → v + √(1 + v²) = cx

Replace v = y/x, so v + √(1 + v²) = (y + √(x² + y²))/x = cx, i.e.

**y + √(x² + y²) = cx²**

#### Alternative form
Isolating y: from √(x² + y²) = cx² − y, squaring gives x² = c²x⁴ − 2cx²y, so

**2cy = c²x² − 1**

(a family of parabolas).

## Check yourself

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

1. x dy - y dx = sqrt(x^2 + y^2) dx is solved by:
   A. y = vx
   B. integrating factor 1/x
   C. z = log y
   D. x = vy
   Answer: A. It is homogeneous. (Key point: Type)

2. After y = vx, the equation separates as:
   A. dv/(1+v^2) = dx/x
   B. dv/sqrt(1+v^2) = dx/x
   C. dv/sqrt(1-v^2) = dx/x
   D. v dv = dx/x
   Answer: B. x^2 dv = x sqrt(1+v^2) dx. (Key point: Reduced equation)

3. Integrating gives:
   A. v - sqrt(1+v^2) = cx
   B. tan^-1 v = ln x + c
   C. v + sqrt(1+v^2) = cx
   D. sin^-1 v = ln x + c
   Answer: C. sinh^-1 v = ln(v + sqrt(1+v^2)). (Key point: Integral)

4. The solution in x and y is:
   A. y + sqrt(x^2 + y^2) = cx^2
   B. y - sqrt(x^2+y^2) = cx^2
   C. y + sqrt(x^2 + y^2) = cx
   D. y^2 = cx^2
   Answer: A. Multiplying v + sqrt(1+v^2) = cx by x. (Key point: Final form)

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