In one line
A function \(f(x,y)\) is homogeneous of degree \(n\) if scaling every variable by \(t\) scales the function by \(t^n\): \(f(tx,ty)=t^nf(x,y)\).
Explained simply
Suppose you double every side of a rectangle. Its perimeter doubles (degree 1) and its area becomes four times (degree 2). The ratio length/breadth does not change at all (degree 0). Functions that react to scaling in such a clean way are called homogeneous, and the power of the scale factor is the degree.
Technically, \(f(x,y)\) is homogeneous of degree \(n\) if \(f(tx,ty)=t^nf(x,y)\) for all \(t>0\). Equivalently, it can be written as \(f(x,y)=x^n\phi\left(\dfrac yx\right)\) or \(y^n\psi\left(\dfrac xy\right)\). The degree \(n\) can be any real number: positive, negative, zero or a fraction. For three variables: \(f(tx,ty,tz)=t^nf(x,y,z)\).
How it works
- Replace \(x\) by \(tx\) and \(y\) by \(ty\) everywhere.
- Try to pull out a single power \(t^n\) from the whole expression.
- If you can, the function is homogeneous of degree \(n\). If different terms leave different powers of \(t\), it is not homogeneous.
- For a polynomial, check that every term has the same total degree.
- For functions like \(\sin^{-1}(\cdot)\) or \(\log(\cdot)\), the function itself is usually not homogeneous, but its inside may be. Then a deduction of Euler's theorem is used instead.
Key formula
\(f(tx,ty)=t^nf(x,y)\quad\Longleftrightarrow\quad f(x,y)=x^n\phi\left(\dfrac yx\right)\). Here \(n\) is the degree, \(t\) a positive scale factor.
Quick example
| Function | After \(x\to tx,\ y\to ty\) | Degree |
|---|---|---|
| \(x^3+3x^2y-y^3\) | \(t^3(\dots)\) | 3 |
| \(\dfrac{x^3+y^3}{x+y}\) | \(\dfrac{t^3}{t}(\dots)=t^2(\dots)\) | 2 |
| \(\sqrt x+\sqrt y\) | \(t^{1/2}(\dots)\) | 1/2 |
| \(\sin^{-1}\dfrac xy\) | unchanged | 0 |
| \(\dfrac{1}{x^2+y^2}\) | \(t^{-2}(\dots)\) | −2 |
| \(x^2+y\) | \(t^2x^2+ty\) | not homogeneous |
Common mistakes
- Deciding the degree from the highest power only; in \(x^2+y\) the terms have different degrees, so it is not homogeneous.
- Thinking \(\sin^{-1}\dfrac{x+y}{\sqrt x+\sqrt y}\) is homogeneous; only \(\sin u\) is (degree 1/2).
- Assuming the degree must be a positive integer.
How MU asks it
Asked as the first step of Euler's theorem questions: "Verify Euler's theorem for \(u=\dfrac{x^3+y^3}{x+y}\)" or as a 3-mark part "State whether \(f\) is homogeneous and find its degree". Always show the \(t\)-substitution and the pulled-out power of \(t\); do not just state the degree.