2 marksMathematicsMultivariable calculusHomogeneous functions and Euler's theoremModerateabout 2 min

IIT JAM 2022 Question 38: Multivariable calculus

Which of the following functions is/are homogeneous?

Select all that apply.

Answer and worked solution

Answer: (A) and (B) and (C)

Scaling $(x,y)\to(tx,ty)$ : in (A), $tx\cot^{-1}(y/x) = t\left[x\cot^{-1}(y/x)\right]$ , homogeneous of degree 1. In (B) every term ( $\sqrt{x/y}$ , $3x/y$ , and the constant 7) is unchanged by scaling, so it is homogeneous of degree 0. In (C) the numerator scales as $t^3$ and the denominator as $t^1$ , so the ratio scales as $t^2$ , homogeneous of degree 2.

In (D) the three terms have total degrees $6$ , $6$ and $7$ ; a mix of degrees means no single $t^r$ factors out, so (D) is not homogeneous. The homogeneous functions are (A), (B) and (C).

The trap: (D)

3x^5 y + 2x^2 y^4 - 3x^3 y^4

A constant such as the $7$ in (B) is homogeneous of degree 0 and does not break homogeneity; what breaks it is mixed term-degrees, as in (D).

Plain-text version

Which of the following functions is/are homogeneous? (A) xcot⁻¹(y/x) (B) √(x/y) + 3x/y + 7 (C) (x³ + y³)/(3x + 4y) (D) 3x⁵ y + 2x² y⁴ - 3x³ y⁴

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What this question tests

This question is on Homogeneous functions and Euler's theorem, part of Multivariable calculus in Mathematics.

Functions that scale in a fixed way when all inputs scale, and the identity this gives.

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