Solve xp-yq=y^2-x^2 in linear pde using multipliers method 

To solve the linear partial differential equation (PDE):

\[
xp - yq = y^2 - x^2
\]

using the **method of multipliers**, we follow these steps:

### Step 1: Rewrite the PDE in Standard Form

The given PDE is:

\[
xp - yq = y^2 - x^2.
\]

This can be rewritten as:

\[
x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y} = y^2 - x^2.
\]

### Step 2: Choose Multipliers

The method of multipliers involves choosing functions \( M(x, y) \) and \( N(x, y) \) such that:

\[
M(x, y) \frac{\partial z}{\partial x} + N(x, y) \frac{\partial z}{\partial y} = M(x, y)(y^2 - x^2).
\]

A common choice for multipliers \( M(x, y) \) and \( N(x, y) \) is to set them proportional to the coefficients of \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \), respectively:

\[
M(x, y) = \frac{1}{x}, \quad N(x, y) = -\frac{1}{y}.
\]

Multiplying the original PDE by these multipliers, we get:

\[
\frac{1}{x} \cdot x \frac{\partial z}{\partial x} - \frac{1}{y} \cdot y \frac{\partial z}{\partial y} = \frac{1}{x} (y^2 - x^2),
\]

which simplifies to:

\[
\frac{\partial z}{\partial x} - \frac{\partial z}{\partial y} = \frac{y^2 - x^2}{x}.
\]

### Step 3: Integrate with Respect to One Variable

Integrate with respect to \( x \):

\[
z(x, y) = \int \left( \frac{\partial z}{\partial x} - \frac{\partial z}{\partial y} \right) \, dx.
\]

Since:

\[
\frac{\partial z}{\partial x} = \frac{y^2 - x^2}{x} + \frac{\partial z}{\partial y},
\]

integrate with respect to \( x \):

\[
z(x, y) = \int \left( \frac{y^2}{x} - x \right) \, dx + \phi(y),
\]

where \( \phi(y) \) is an arbitrary function of \( y \). 

### Step 4: Perform the Integration

Now, integrate:

\[
z(x, y) = \int \left( \frac{y^2}{x} - x \right) \, dx + \phi(y).
\]

This yields:

\[
z(x, y) = y^2 \ln |x| - \frac{x^2}{2} + \phi(y).
\]

### Step 5: Determine the General Solution

The general solution to the PDE is:

\[
z(x, y) = y^2 \ln |x| - \frac{x^2}{2} + \phi(y),
\]

where \( \phi(y) \) is an arbitrary function of \( y \).

### Conclusion

The solution to the linear PDE using the multipliers method is:

\[
\boxed{z(x, y) = y^2 \ln |x| - \frac{x^2}{2} + \phi(y)},
\]

where \( \phi(y) \) is any differentiable function of \( y \).

Would you like more details or have any questions?

### Related Questions:

1. How to choose appropriate multipliers for more complex PDEs?
2. What are the advantages of the multipliers method compared to other methods?
3. Can the multipliers method be applied to non-linear PDEs?
4. How to determine the function \( \phi(y) \) if additional boundary conditions are given?
5. What is the geometric interpretation of the solution to this PDE?

**Tip:** When solving PDEs, always consider the characteristics of the differential equation to choose the most efficient solving method.

Solve xp - yq = y^2 - x^2 in linear pde using multipliers method

Discover how to solve the linear partial differential equation xp - yq = y^2 - x^2 using the method of multipliers. This step-by-step guide covers the selection of multipliers, integration, and general solution, suitable for advanced university-level mathematics.

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