Verify Euler’s theorem for the function u = f (x, y) = ax + 2hxy + by .

Euler’s theorem states that for a homogeneous function of degree \( n \), the following relation holds:

\[
x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u
\]

Here, we need to verify Euler's theorem for the function:

\[
u(x, y) = ax + 2hxy + by
\]

To do this, we will:

1. **Check if the function is homogeneous**: A function is homogeneous if there exists a degree \( n \) such that scaling all variables by a factor \( \lambda \) results in scaling the function by \( \lambda^n \). This means:

\[
f(\lambda x, \lambda y) = \lambda^n f(x, y)
\]

2. **Compute the partial derivatives** \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \).

3. **Apply Euler’s theorem** by checking if \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = nu \) holds.

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### Step 1: Check if \( u(x, y) \) is homogeneous

Let's scale \( x \) and \( y \) by \( \lambda \), i.e., substitute \( x = \lambda x \) and \( y = \lambda y \) into \( u(x, y) \):

\[
u(\lambda x, \lambda y) = a(\lambda x) + 2h(\lambda x)(\lambda y) + b(\lambda y)
= \lambda a x + 2h \lambda^2 xy + \lambda b y
\]

Factoring out \( \lambda \):

\[
u(\lambda x, \lambda y) = \lambda(ax + by) + \lambda^2(2hxy)
\]

We can see that \( u(\lambda x, \lambda y) \) is not proportional to a single power of \( \lambda \), so it is **not homogeneous**. This means that Euler’s theorem, which applies to homogeneous functions, does **not apply** to this function.

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Since the function is not homogeneous, Euler's theorem is not valid for this function, and the verification stops here.

Would you like more details or clarification on any part?

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Here are 5 follow-up questions related to this concept:

1. How can you identify the degree of a homogeneous function?
2. What happens if a function is homogeneous of degree zero?
3. Can Euler's theorem be applied to functions with mixed degree terms?
4. How does the scaling factor influence homogeneous functions?
5. Can a non-homogeneous function become homogeneous under a different transformation?

**Tip**: Homogeneous functions play a key role in fields like economics, physics, and differential equations where scaling properties are important.

Verify Euler’s theorem for the function u = f (x, y) = ax + 2hxy + by.

Explore how Euler's theorem applies to the function u(x, y) = ax + 2hxy + by. This detailed explanation covers the identification of homogeneous functions, the computation of partial derivatives, and the application of Euler’s theorem for homogeneous functions of degree n. Ideal for college-level students.

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