https://file.aiquickdraw.com/mathbot/file/1729306891913rn31bk0d.jpg

The given image contains a partial differential equation and a problem statement. Upon reading and interpreting the text, it says:

\[
\frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0 \quad (1)
\]

The task appears to be:  
*"If \( u = x \), then show that (1) holds."*

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### Solution

1. **Problem Overview**:  
We are given a function \( u = x \) and asked to verify whether it satisfies the equation:

   \[
   \frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0.
   \]

2. **Partial Derivatives**:
   - \( u = x \)
     - \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x) = 1 \).
     - \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(x) = 0 \) (since \( u = x \) does not depend on \( y \)).

3. **Substitute into the Equation**:
   Substituting the partial derivatives into the equation \( \frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0 \):

   \[
   1 + 3y \cdot 0 = 1 \neq 0.
   \]

4. **Conclusion**:
   The equation \( \frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0 \) does **not** hold for \( u = x \).

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Would you like to explore further details or discuss related concepts?  

### 5 Follow-up Questions:
1. How do partial differential equations differ from ordinary differential equations?
2. What are the general methods to solve first-order PDEs like the given one?
3. Could a different form of \( u(x, y) \) satisfy the given PDE?
4. What is the significance of boundary conditions in PDEs?
5. How do characteristics help in solving PDEs analytically?

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**Tip:** When dealing with PDEs, always check if the function you're testing satisfies all the terms—sometimes the solution is trivial, but other times it requires transformations.

If u = x, then show that \( \frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0 \).

Explore how to verify the partial differential equation \( \frac{\partial u}{\partial x} + 3y \frac{\partial u}{\partial y} = 0 \) given that u = x. The solution involves calculating partial derivatives and substituting into the equation. Suitable for undergraduate students learning PDEs.

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