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 $u(x, y)$:

= \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. --- 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? --- 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.