To solve the problem stated in the image, I'll break down the given problem step by step.

Problem:

You are asked to prove:

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

Also, there is a second part asking to evaluate $x \frac{\partial y}{\partial x} + y \frac{\partial y}{\partial y}$ under some conditions.

Step 1: Analyze Given Functions and Partial Derivatives

From the image, the equation seems to involve partial derivatives of $y$ with respect to $x$ and $y$. This implies that $y$ is dependent on $x$ and another variable. Let us rewrite the key components for clarity.

Function Definitions:

Assume the relations:

• $y = f(x, y_2)$, where $y_2$ is a variable and relates to $x$ through other terms.
• There might also be trigonometric relations, as suggested by $\cot u$.

Step 2: Proof Setup

We need to calculate and verify the left-hand side:

$x \frac{\partial y}{\partial x} + y \frac{\partial y}{\partial y}$

Partial Derivatives:

1. $\frac{\partial y}{\partial x}$ depends on how $y$ varies with $x$, treating all other variables constant.
2. $\frac{\partial y}{\partial y}$ measures how $y$ depends on itself, likely contributing a proportional relationship.

Right-Hand Side:

The right-hand side $2 \cot u$ suggests there’s a trigonometric function embedded within the equations.

Step 3: Solve and Verify the Second Part

The second part of the question involves explicitly evaluating $x \frac{\partial y}{\partial x} + y \frac{\partial y}{\partial y}$ under given constraints.

Can you confirm the equation setup or provide clarification for any missing terms? I’ll use these assumptions to compute the detailed solution.