Q.Which of the following is not a homogeneous function of $x$ and $y$ ?
(A) $y^2 - xy$
(B) $x - 3y$
(C) $\sin^2 \frac{y}{x} + \frac{y}{x}$
(D) $\tan x - \sec y$

Concept: Homogeneous Function Degree — a function $f(x, y)$ is homogeneous of degree $n$ if $f(tx, ty) = t^n f(x, y)$ for all $t > 0$.

Step 1: Check each option by replacing $x$ with $tx$ and $y$ with $ty$.

• (A) $(ty)^2 - (tx)(ty) = t^2(y^2 - xy)$ → degree 2.
• (B) $tx - 3ty = t(x - 3y)$ → degree 1.
• (C) $\sin^2\frac{ty}{tx} + \frac{ty}{tx} = \sin^2\frac{y}{x} + \frac{y}{x}$ → degree 0. …

The idea of a homogeneous function is simple: if you scale both inputs by the same factor $t$, the output scales by $t$ raised to some fixed power $n$. That power $n$ is called the degree of homogeneity. This property is extremely useful in differential equations and economics — whenever you see a function where every term has the same total exponent, or where the function depends only on ratios like $y/x$, you are likely looking at a homogeneous function.

Let’s test each option systematically.

1. Option (A): $f(x, y) = y^2 - xy$ Replace $x$ with $tx$ and $y$ with $ty$:

This is homogeneous of degree 2. Every term is degree 2 (since $y^2$ is degree 2, and $xy$ is also degree 2). So (A) is homogeneous.

2. Option (B): $f(x, y) = x - 3y$

This is homogeneous of degree 1. Both terms are linear. So (B) is homogeneous.

3. Option (C): $f(x, y) = \sin^2 \frac{y}{x} + \frac{y}{x}$ Here the function depends only on the ratio $y/x$. Replace $x$ with $tx$ and $y$ with $ty$:

Notice that $f(tx, ty) = t^0 f(x, y)$ because $f$ does not change at all — it is homogeneous of degree 0. Any function that depends only on the ratio $y/x$ (or $x/y$) is automatically homogeneous of degree 0. So (C) is homogeneous.

4. Option (D): $f(x, y) = \tan x - \sec y$ Replace $x$ with $tx$ and $y$ with $ty$: $f(tx, ty) = \tan(tx) - \sec(ty).$ …

Method: Testing whether a function is homogeneous

Use this to decide if a function $f(x,y)$ is homogeneous, and of what degree — the property that lets a DE be solved by the substitution $y=vx$.

Steps

Step 1: Replace $x\to tx$ and $y\to ty$

Substitute both variables scaled by the same factor $t$ and simplify.

Step 2: Try to pull out a common power of $t$

$f(x,y)$ is homogeneous of degree $n$ if

for every $t$. Every term must carry the same total power of $t$.

Step 3: Flag the failures …

Common Mistakes

Mistake 1: Assuming $\sin^2\frac{y}{x}+\frac{y}{x}$ is not homogeneous because of the $\sin$

Why it's wrong: any function of the ratio $\frac{y}{x}$ is homogeneous of degree $0$, since scaling $x,y$ by $t$ leaves $\frac{y}{x}$ unchanged. Correct approach: it is homogeneous (degree $0$).

Mistake 2: Not testing every option with $x\to tx,\ y\to ty$ …