Homogeneous functions
Before you start: you need partial differentiation.
\(f(x, y)\) is homogeneous of degree \(n\) if, for every \(t > 0\), \[f(tx, ty) = t^n f(x, y)\]
Equivalent form
\(f(x, y) = x^n\,\phi\!\left(\dfrac{y}{x}\right)\) for some function \(\phi\).
Quick test for polynomials: every term has the same total degree. \(x^3 + 3x^2y + y^3\) is homogeneous of degree 3, because every term has degree 3. For a fraction, the degree is (degree of the top) − (degree of the bottom), provided both are homogeneous.
| Function | Degree |
|---|---|
| \(x^2 + xy + y^2\) | \(2\) |
| \(\dfrac{x^3 + y^3}{x + y}\) | \(3 - 1 = 2\) |
| \(\dfrac{x + y}{\sqrt x + \sqrt y}\) | \(1 - \tfrac{1}{2} = \tfrac{1}{2}\) |
| \(\tan^{-1}\dfrac{y}{x}\) | \(0\) |
| \(x^2 + y\) | not homogeneous |
Euler's theorem
If \(u\) is a homogeneous function of \(x\) and \(y\) of degree \(n\), then \[x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = n\,u\]
Three variables
\(x u_x + y u_y + z u_z = n\,u\)
Second-order form
\[x^2 u_{xx} + 2xy\,u_{xy} + y^2 u_{yy} = n(n - 1)\,u\]
Proof
Differentiate \(u(tx, ty) = t^n u(x, y)\) with respect to \(t\), using the chain rule: \(x\,u_x(tx, ty) + y\,u_y(tx, ty) = n\,t^{n-1}u(x, y)\). Now put \(t = 1\). \(\blacksquare\)
The extension exam questions love
Many exam functions are not homogeneous themselves, for example \(u = \sin^{-1}\dfrac{x^2 + y^2}{x + y}\). But some function of them is: here \(\sin u = \dfrac{x^2 + y^2}{x + y}\) is homogeneous of degree 1.
If \(F(u)\) is homogeneous of degree \(n\) in \(x\) and \(y\), then \[x u_x + y u_y = n\,\frac{F(u)}{F'(u)} = g(u)\]
Second-order form
\[x^2 u_{xx} + 2xy\,u_{xy} + y^2 u_{yy} = g(u)\big[g'(u) - 1\big]\]
Why it works: apply Euler's theorem to \(F(u)\), whose partial derivatives are \(F'(u)\,u_x\) and \(F'(u)\,u_y\). That gives \(F'(u)(x u_x + y u_y) = n F(u)\).
Solved examples
Example 1: Verifying the theorem
EasyVerify Euler's theorem for \(u = x^3 + 3x^2y + y^3\).
- Every term has degree 3, so \(n = 3\).
- \(u_x = 3x^2 + 6xy\) and \(u_y = 3x^2 + 3y^2\).
- \(x u_x + y u_y = 3x^3 + 6x^2y + 3x^2y + 3y^3 = 3x^3 + 9x^2y + 3y^3 = 3u\). ✓
AnswerVerified: \(x u_x + y u_y = 3u\).
Example 2: A fraction
EasyIf \(u = \dfrac{x^2 + y^2}{x + y}\), find \(x u_x + y u_y\).
- Degree \(= 2 - 1 = 1\).
- By Euler's theorem, \(x u_x + y u_y = 1\cdot u\).
Answer\(x u_x + y u_y = u\)
Example 3: Inverse sine
MediumIf \(u = \sin^{-1}\dfrac{x^2 + y^2}{x + y}\), show that \(x u_x + y u_y = \tan u\).
- \(\sin u = \dfrac{x^2 + y^2}{x + y}\) is homogeneous of degree 1. So take \(F(u) = \sin u\), \(n = 1\).
- \(x u_x + y u_y = n\dfrac{F(u)}{F'(u)} = \dfrac{\sin u}{\cos u}\).
Answer\(x u_x + y u_y = \tan u\)
Example 4: Inverse tangent
MediumIf \(u = \tan^{-1}\dfrac{x^3 + y^3}{x - y}\), show that \(x u_x + y u_y = \sin 2u\).
- \(\tan u = \dfrac{x^3 + y^3}{x - y}\) is homogeneous of degree \(3 - 1 = 2\).
- \(x u_x + y u_y = 2\dfrac{\tan u}{\sec^2 u} = 2\sin u\cos u\).
Answer\(x u_x + y u_y = \sin 2u\)
Example 5: Logarithm
MediumIf \(u = \log\dfrac{x^4 + y^4}{x + y}\), find \(x u_x + y u_y\).
- \(e^u = \dfrac{x^4 + y^4}{x + y}\) is homogeneous of degree 3. Take \(F(u) = e^u\).
- \(x u_x + y u_y = 3\dfrac{e^u}{e^u}\).
Answer\(x u_x + y u_y = 3\)
Example 6: Second order
Exam levelIf \(u = \sin^{-1}\dfrac{x + y}{\sqrt x + \sqrt y}\), show that \(x u_x + y u_y = \tfrac{1}{2}\tan u\) and \(x^2 u_{xx} + 2xy\,u_{xy} + y^2 u_{yy} = -\dfrac{\sin u\cos 2u}{4\cos^3 u}\).
- \(\sin u\) is homogeneous of degree \(\tfrac{1}{2}\), so \(g(u) = \tfrac{1}{2}\dfrac{\sin u}{\cos u} = \tfrac{1}{2}\tan u\). That's the first result.
- \(g'(u) = \tfrac{1}{2}\sec^2 u\), so \(g'(u) - 1 = \dfrac{1 - 2\cos^2 u}{2\cos^2 u} = -\dfrac{\cos 2u}{2\cos^2 u}\).
- \(g(u)\big[g'(u) - 1\big] = \dfrac{\sin u}{2\cos u}\cdot\left(-\dfrac{\cos 2u}{2\cos^2 u}\right)\).
Answer\(x^2 u_{xx} + 2xy\,u_{xy} + y^2 u_{yy} = -\dfrac{\sin u\cos 2u}{4\cos^3 u}\)
When \(u\) is wrapped in \(\sin^{-1}\), \(\tan^{-1}\) or \(\log\), undo the wrapper first: \(\sin u\), \(\tan u\) or \(e^u\) is usually the homogeneous function. Then the answer is just \(n\dfrac{F(u)}{F'(u)}\).
Common mistakes
1. Applying the theorem to a non-homogeneous \(u\)
\(u = \sin^{-1}\dfrac{x^2 + y^2}{x + y}\) is not homogeneous, so \(x u_x + y u_y \ne u\). Use the extension with \(F(u) = \sin u\).
2. Getting the degree of a fraction wrong
Degree of \(\dfrac{\text{top}}{\text{bottom}}\) = degree of the top minus degree of the bottom. Square roots count as degree \(\tfrac{1}{2}\).
3. Using \(n(n - 1)u\) for the extension
The second-order result \(n(n - 1)u\) is only for homogeneous \(u\). For the extension, use \(g(u)[g'(u) - 1]\).
Practice questions
Try each one on paper before you open the answer.
Q1What is the degree of \(\dfrac{x^3 + y^3}{x^2 + y^2}\)?
\(3 - 2 = 1\).
Q2If \(u = x^2 y + x y^2\), find \(x u_x + y u_y\).
Degree 3, so \(3u\).
Q3If \(u = \tan^{-1}\dfrac{y}{x}\), find \(x u_x + y u_y\).
Degree 0, so \(0\).
Q4If \(u = \log\dfrac{x^2 + y^2}{x + y}\), find \(x u_x + y u_y\).
\(e^u\) has degree 1, so \(x u_x + y u_y = 1\).
Q5If \(u = \cos^{-1}\dfrac{x + y}{\sqrt x + \sqrt y}\), show that \(x u_x + y u_y = -\tfrac{1}{2}\cot u\).
\(\cos u\) has degree \(\tfrac{1}{2}\): \(\tfrac{1}{2}\dfrac{\cos u}{-\sin u} = -\tfrac{1}{2}\cot u\).
Q6For \(u = x^3 + y^3\), verify that \(x^2 u_{xx} + 2xy\,u_{xy} + y^2 u_{yy} = 6u\).
\(u_{xx} = 6x\), \(u_{xy} = 0\), \(u_{yy} = 6y\). The left side is \(6x^3 + 6y^3 = 6u = 3\cdot 2\cdot u\). ✓
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Frequently asked questions
What is a homogeneous function?
One where scaling every variable by \(t\) scales the function by \(t^n\): \(f(tx, ty) = t^n f(x, y)\). The number \(n\) is its degree.
What does Euler's theorem say?
For a homogeneous function of degree \(n\), \(x u_x + y u_y = n u\).
What if u itself is not homogeneous?
Look for a function \(F(u)\) that is, such as \(\sin u\), \(\tan u\) or \(e^u\). Then \(x u_x + y u_y = n\dfrac{F(u)}{F'(u)}\).
Related lessons
← All derivatives lessons
Vipul Sir has taught mathematics for over 15 years to HSC, CBSE, ICSE, IGCSE/IB, Diploma, Engineering and BSc students at VVS Classes in Goregaon and Vile Parle, Mumbai. He teaches every class himself, concepts first and exam technique second.