Euler’s theorem on homogeneous functions is one of the short results in the partial differentiation part of VTU Engineering Mathematics 1 (BMATS101 under the 2022 scheme, 18MAT11 under 2018). It is brief, but it appears in questions that test whether you can handle partial derivatives without writing every step, and students who skip it often lose marks on the verification and second-order parts. This guide states the theorem, explains what a homogeneous function is, and works through examples in full.
What a Homogeneous Function Is
A function f(x, y) is homogeneous of degree n if, when you multiply both x and y by the same number t, the function is multiplied by tⁿ:
f(tx, ty) = tⁿ f(x, y)
Test this on a simple case. Take f(x, y) = x² + xy. Then f(tx, ty) = t²x² + t²xy = t²(x² + xy). So f is homogeneous of degree 2. The degree is the power of t that comes out.
Not every function is homogeneous. The function x² + y is not, because the term y scales like t, not t². A function is homogeneous only if all its terms have the same total degree.
Euler’s Theorem (First Order)
If u = f(x, y) is homogeneous of degree n, then:
x ∂u/∂x + y ∂u/∂y = n u
This is a clean statement, and it is the form most exam questions use. The left side combines the partial derivatives with the variables, and the right side is a constant multiple of u. You can use it to check a calculation or to simplify an expression without computing every derivative explicitly.
Worked Example 1: Verifying the Theorem for u = x³ + 3x²y + y³
Each term has total degree 3, so u is homogeneous of degree 3. Compute the partial derivatives:
- ∂u/∂x = 3x² + 6xy
- ∂u/∂y = 3x² + 3y²
Now form the left side of Euler’s theorem:
x ∂u/∂x + y ∂u/∂y = x(3x² + 6xy) + y(3x² + 3y²)
= 3x³ + 6x²y + 3x²y + 3y³ = 3x³ + 9x²y + 3y³
The right side is 3u = 3(x³ + 3x²y + y³) = 3x³ + 9x²y + 3y³. The two sides match, so the theorem holds.
Verification like this is a common exam request. If a question asks you to verify Euler’s theorem, you should show both sides separately and show that they are equal.
Worked Example 2: A Degree-Zero Function
Let u = x/y. Each term has total degree 0 (x/y scales as t/t = 1), so u is homogeneous of degree 0.
- ∂u/∂x = 1/y
- ∂u/∂y = −x/y²
Then x ∂u/∂x + y ∂u/∂y = x/y − x/y = 0. Since n = 0, the right side is 0 · u = 0. The theorem holds again.
Degree-zero functions are often used in questions about ratios. The result tells you that such a function is unchanged when x and y are scaled together, which is why its Euler sum vanishes.
Worked Example 3: The Second-Order Form
Euler’s theorem has a second-order version that appears in many VTU questions:
x² ∂²u/∂x² + 2xy ∂²u/∂x∂y + y² ∂²u/∂y² = n(n−1) u
Check it with the same function, u = x³ + 3x²y + y³, which has n = 3. The second derivatives are:
- ∂²u/∂x² = 6x + 6y
- ∂²u/∂x∂y = 6x
- ∂²u/∂y² = 6y
Substitute into the left side:
x²(6x + 6y) + 2xy(6x) + y²(6y) = 6x³ + 6x²y + 12x²y + 6y³ = 6x³ + 18x²y + 6y³
The right side is n(n−1)u = 3 · 2 · u = 6u = 6x³ + 18x²y + 6y³. The two sides match.
The second-order form is the one students most often get wrong, because the coefficient 2 on the mixed term is easy to forget. Write the coefficient out every time.
How to Use Euler’s Theorem in an Exam
- Confirm the function is homogeneous. Check that every term has the same total degree, and state that degree n.
- Use the first-order form to simplify. If a question asks for x u_x + y u_y, write n u directly without computing the derivatives.
- Use the second-order form when second derivatives are involved. Remember the n(n−1) factor on the right.
- Verify by direct computation when asked. Show both sides separately and confirm they match.
Common Mistakes That Cost Marks
Mistake 1: Applying the theorem to a non-homogeneous function. Check the degree of every term before writing n. The function x² + y is not homogeneous, and Euler’s theorem does not apply to it.
Mistake 2: Using the wrong degree. For x³/y, the degree is 3 − 1 = 2, not 3. Subtract the degree of the denominator.
Mistake 3: Forgetting the coefficient 2 in the second-order form. The mixed term carries a factor of 2. Missing it breaks the identity.
Mistake 4: Writing n(n−1) when the degree is 1 or 0. For n = 1, the second-order right side is 0. For n = 0 it is also 0. Check the value of n before substituting.
Practice Questions
- Verify Euler’s theorem for u = x² + 2xy + 3y² (degree 2).
- If u = x² y/(x + y), show that x u_x + y u_y = 2u. (Hint: check the degree first.)
- For u = x/(x² + y²) show that x u_x + y u_y = −u. (Hint: the function is homogeneous of degree −1.)
Checks: (1) u_x = 2x + 2y, u_y = 2x + 6y. x u_x + y u_y = 2x² + 2xy + 2xy + 6y² = 2x² + 4xy + 6y² = 2u ✓. (2) Degree is 2 + 1 − 1 = 2, so n = 2 ✓. (3) Degree is 1 − 2 = −1, so the right side is −u. Verify by computing the derivatives directly.
A Worked Example with a Non-Polynomial Function
Euler’s theorem is not limited to polynomials. Take u = x e^(y/x). Each term scales as follows: replacing x by tx and y by ty gives tx · e^(ty/tx) = t · x e^(y/x), so u is homogeneous of degree 1.
Compute the partial derivatives. For the first:
∂u/∂x = e^(y/x) + x · e^(y/x) · (−y/x²) = e^(y/x) − (y/x) e^(y/x)
For the second:
∂u/∂y = x · e^(y/x) · (1/x) = e^(y/x)
Now form the Euler sum:
x ∂u/∂x + y ∂u/∂y = x e^(y/x) − y e^(y/x) + y e^(y/x) = x e^(y/x)
The right side is 1 · u = x e^(y/x), so the theorem holds. The exponential term contributed nothing extra, because the degree-1 scaling was built into the structure of u. This is the reason the theorem is useful: once you know the degree, you can check the answer without expanding every derivative.
Why the Degree Must Be Confirmed First
The most common error in Euler questions is applying the theorem without confirming homogeneity. For functions such as sin(x/y) or x + y², the scaling test fails, and the theorem gives a wrong result. Before writing n, substitute tx and ty and check that the function is multiplied by a single power of t. If it is not, the theorem does not apply, and you must compute the partial derivatives directly.
Connecting This Topic to the Full Module
Euler’s theorem belongs to the partial differentiation part of VTU Engineering Mathematics 1, alongside Jacobians and maxima and minima of functions of two variables. Seeing how these topics fit together helps with the mixed questions that combine them. The full outline, with the topics most often tested, is in our VTU Engineering Maths 1 Notes — Complete Module-Wise Guide (BMATS101).
Frequently Asked Questions
Q: Does Euler’s theorem work for functions of three variables?
Yes. The first-order form extends by adding a term z ∂u/∂z. Your syllabus may only use two variables, so check your course notes.
Q: How do I tell quickly whether a function is homogeneous?
Add the powers in each term. If the total degree is the same for every term, the function is homogeneous of that degree.
Q: Where does this fit in the module?
It is taught alongside the partial differentiation and maxima-minima material. The full module outline is in our VTU Engineering Maths 1 notes guide.
For more solved partial differentiation problems and formula sheets, see the Engineering Math Survival Kit. To ask about batches, use the contact page.