---
title: "Euler's theorem verified for xⁿ tan⁻¹(y/x)"
subject: "Mathematics"
chapter: 1
key: "semester_2/mathematics:euler-s-theorem-verified-x"
source: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-verified-x
site: BCT Engineering
---

# Euler's theorem verified for xⁿ tan⁻¹(y/x)

Mathematics · Chapter 1: Calculus of Two or more variables

- Asked in: 2073 Bhadra, 5 marks
- Video: [Verification of Eulers theorem problem 1](https://www.youtube.com/watch?v=PXyH6lzabBI) — TEAM Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/euler-s-theorem-verified-x
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-1

## Question

State Euler's Theorem for a homogeneous function of two independent variables and verify it for the function u = xⁿ tan⁻¹(y/x).

## Short answer

Euler's theorem states that for a homogeneous function f of degree n, x f_x + y f_y = n f. For u = x^n arctan(y/x), it is homogeneous degree n, and direct calculation gives x u_x + y u_y = n u, verifying the theorem.

## Answer

#### Euler's theorem
If f(x, y) is homogeneous of degree *n* (f(tx, ty) = tⁿ f(x, y)) and has continuous first partial derivatives, then

x ∂f/∂x + y ∂f/∂y = n f.

#### Verification for u = xⁿ tan⁻¹(y/x)
Degree: u(tx, ty) = tⁿ xⁿ tan⁻¹(ty/tx) = tⁿ u, so u is homogeneous of degree n. Euler's theorem predicts x u_x + y u_y = n u.

Partial derivatives (using d/dx tan⁻¹(y/x) = −y/(x² + y²) and d/dy tan⁻¹(y/x) = x/(x² + y²)):

```text
∂u/∂x = n xⁿ⁻¹ tan⁻¹(y/x) + xⁿ ( −y/(x² + y²) )
∂u/∂y = xⁿ · x/(x² + y²) = xⁿ⁺¹/(x² + y²)
```

Then

```text
x ∂u/∂x = n xⁿ tan⁻¹(y/x) − xⁿ⁺¹ y/(x² + y²)
y ∂u/∂y = xⁿ⁺¹ y/(x² + y²)
```

Adding, the fractions cancel:

```text
x ∂u/∂x + y ∂u/∂y = n xⁿ tan⁻¹(y/x) = n u
```

This verifies Euler's theorem. ∎

## Check yourself

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

1. What is the degree of homogeneity of u = xⁿ tan⁻¹(y/x)?
   A. 0
   B. n+1
   C. n−1
   D. n
   Answer: D. u(tx, ty) = tⁿ xⁿ tan⁻¹(y/x) = tⁿ u. (Key point: Degree)

2. What is ∂/∂y of tan⁻¹(y/x)?
   A. −y/(x²+y²)
   B. x/(x²+y²)
   C. y/(x²+y²)
   D. 1/(x²+y²)
   Answer: B. d/dy tan⁻¹(y/x) = (1/x)/(1+y²/x²) = x/(x²+y²). (Key point: Derivative)

3. After forming x u_x + y u_y, what happens to the fraction terms?
   A. They add to give 2
   B. They cancel, leaving n xⁿ tan⁻¹(y/x) = n u
   C. They are squared
   D. They give xⁿ
   Answer: B. x u_x has -xⁿ⁺¹y/(x²+y²) and y u_y has +xⁿ⁺¹y/(x²+y²); they cancel. (Key point: Verification)

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