Evaluating Reason (R): Homogeneity of the Function

Let the given function be $u = xy f\left(\frac{y}{x}\right)$. To check if it is homogeneous, we replace $x$ with $tx$ and $y$ with $ty$.

$ u(tx, ty) = (tx)(ty) f\left(\frac{ty}{tx}\right) $ $ u(tx, ty) = t^2 xy f\left(\frac{y}{x}\right) $ Since $u(x, y) = xy f\left(\frac{y}{x}\right)$, we have: $ u(tx, ty) = t^2 u(x, y) $ This confirms that the function $u$ is homogeneous of degree 2 in $x$ and $y$. Therefore, Reason (R) is correct.

Evaluating Assertion (A): Applying Euler's Theorem

Euler's theorem states that if a function $u(x, y)$ is homogeneous of degree $n$, then it satisfies the partial differential equation:

$ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = nu $

From the evaluation of Reason (R), we established that the given function $u$ is homogeneous of degree $n=2$. Applying Euler's theorem with $n=2$, we get:

$ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2u $

This matches exactly what is stated in Assertion (A). Therefore, Assertion (A) is correct.

Relationship between Assertion (A) and Reason (R)

Reason (R) correctly identifies the function $u$ as homogeneous of degree 2. Assertion (A) is a direct application of Euler's theorem based on this property of homogeneity. Thus, Reason (R) provides the correct explanation for Assertion (A).

Both statements are correct, and Reason (R) is the correct explanation of Assertion (A).