Solve using Lagrange method for partial differential equations xp- yq -xy=0 

To solve the partial differential equation (PDE) \( xp - yq - xy = 0 \) using the **Lagrange method**, we follow these steps:

### Step 1: Recognize the standard form of the PDE
We write the given PDE in terms of partial derivatives:
\[
xp - yq - xy = 0
\]
Here:
- \( p = \frac{\partial z}{\partial x} \)
- \( q = \frac{\partial z}{\partial y} \)

Thus, the equation becomes:
\[
x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y} - xy = 0
\]

### Step 2: Write the Lagrange auxiliary equations
The general form of a first-order PDE for Lagrange's method is:
\[
\frac{dx}{A} = \frac{dy}{B} = \frac{dz}{C}
\]
From the PDE, we identify the coefficients of \( p \) and \( q \) and rewrite the equation as:
\[
\frac{dx}{x} = \frac{dy}{-y} = \frac{dz}{xy}
\]

### Step 3: Solve the system of Lagrange equations

#### 1. Solve \( \frac{dx}{x} = \frac{dy}{-y} \)
We integrate both sides separately:
\[
\int \frac{dx}{x} = \int \frac{dy}{-y}
\]
This yields:
\[
\ln |x| = -\ln |y| + \ln C_1
\]
Simplifying:
\[
\ln |x| + \ln |y| = \ln C_1
\]
\[
\ln |xy| = \ln C_1
\]
Thus, we get the first relation:
\[
xy = C_1
\]

#### 2. Solve \( \frac{dx}{x} = \frac{dz}{xy} \)
From this, we get:
\[
\frac{dz}{xy} = \frac{dx}{x}
\]
Integrating both sides:
\[
\int \frac{dz}{xy} = \int \frac{dx}{x}
\]
Since \( xy = C_1 \), this reduces to:
\[
\frac{1}{C_1} \int dz = \int \frac{dx}{x}
\]
\[
\frac{z}{C_1} = \ln |x| + \ln C_2
\]
Multiplying by \( C_1 \):
\[
z = C_1 \ln |x| + C_3
\]

### Step 4: General solution
Thus, the general solution of the PDE is:
\[
z = C_1 \ln |x| + C_3
\]
where \( C_1 \) is a function of \( xy \) from the first relation, i.e., \( C_1 = f(xy) \).

### Final General Solution:
\[
z = f(xy) \ln |x| + C
\]

Do you want more details or have any questions?

Here are five related questions to expand this concept:
1. How does the Lagrange method differ from other methods of solving PDEs?
2. What kind of problems are best suited for the Lagrange method?
3. Can the method be applied if the PDE contains higher-order derivatives?
4. How would we approach this if the PDE involved non-linear terms?
5. What are the limitations of the Lagrange method?

**Tip:** Always check the form of the given PDE and see if it can be expressed in the first-order form for Lagrange’s method to apply smoothly.

Solve using Lagrange method for partial differential equations xp - yq - xy = 0

This tutorial demonstrates solving the partial differential equation xp - yq - xy = 0 using the Lagrange method. We explore key steps, such as writing the Lagrange auxiliary equations and solving them to find the general solution. Suitable for advanced university students studying differential equations.

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