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The image contains two problems related to Partial Differential Equations (PDEs). Here's a breakdown of each problem:

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### 3. Create a PDE with \( u = u(x,y) \) as the solution, where \( f \) and \( g \) are arbitrary functions:
1. \( u = x f\left(\frac{y}{x}\right) \)
2. \( u = f(x^2 - xy) + g(x^2 + xy) \)  \(\quad (\text{with } g \text{ constant})\)
3. \( \left( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \right) = 0 \)

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### 4. Solve the following PDEs with the given conditions:
1. \( u_x + u_y = u^2, \quad u(x, 0) = 0 \)
2. \( x u_x + y u_y = x^2, \quad u(1, y) = y^2 \)
3. \( x^2 u_{xy} + u_y = x \ln y, \quad u(x, y) = \frac{1}{x^2} \ln x, \quad u(x, 3x) = x^2 \)
4. \( u_x + u_y = 0, \quad u(x, 0) = 0, \quad u(\sqrt{2}, 0) = \sqrt{2}, \quad u(x, 0) = 0 \text{ for } x > 0, y > 0 \)

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Would you like me to solve a specific part of the problem, or provide further explanations on any of the questions? Let me know if you need detailed solutions!

Here are some related questions:
1. What is a general approach to solving first-order PDEs like \( u_x + u_y = u^2 \)?
2. How can you classify second-order PDEs based on their characteristics?
3. How would the method of separation of variables apply to PDEs like the ones in question 4?
4. Can you explain how the method of characteristics is used for first-order PDEs?
5. What is the significance of boundary conditions in solving PDEs?

**Tip:** When solving PDEs, it's important to first understand the type of the equation (e.g., elliptic, parabolic, or hyperbolic), as different methods apply depending on the classification.

Make a PDE with u = u(x,y) as the solution. Solve the PDE with given conditions.

Explore the solution process for partial differential equations with conditions. This problem covers first-order and second-order PDEs using methods like separation of variables and the method of characteristics. Suitable for undergraduate or graduate-level students.

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