## Solving the Partial Differential Equation (x^2+y^2)p+2xyq=(x+y)z

This article explores the solution of the given partial differential equation (PDE):

**(x^2+y^2)p+2xyq=(x+y)z**

where 'p' represents the partial derivative of 'z' with respect to 'x' (∂z/∂x), and 'q' represents the partial derivative of 'z' with respect to 'y' (∂z/∂y). This equation falls under the category of **non-linear first-order PDEs**.

### Identifying the Type

This PDE is **non-linear** due to the presence of the product term '2xyq'. It's also a **first-order** PDE because the highest order derivative present is the first derivative (p and q).

### Solution Approach

A common approach to solving non-linear first-order PDEs is using the **Lagrange's method** or the **Charpit's method**.

**1. Lagrange's Method:**

This method involves finding two independent solutions (u and v) of the auxiliary equations:

```
dx / (x^2+y^2) = dy / (2xy) = dz / (x+y)z
```

Solving the first two equations, we get:

```
(x^2 + y^2)dy = 2xydx
```

This can be solved by separation of variables, leading to:

```
y^2 / x^2 = C1
```

where C1 is an arbitrary constant.

Similarly, solving the second and third equations, we get:

```
(x+y)dz = z(2xydy)
```

This also can be solved by separation of variables, resulting in:

```
z / (x^2 * y) = C2
```

where C2 is another arbitrary constant.

Finally, we can express the general solution of the PDE as:

```
F(C1, C2) = 0
```

where F is an arbitrary function. This means the solution can be expressed as:

```
F( y^2 / x^2, z / (x^2 * y)) = 0
```

**2. Charpit's Method:**

This method involves finding a complete integral of the PDE. This can be achieved by introducing auxiliary equations:

```
dp = (x+y)z * ds
dq = -(x^2+y^2)z * ds
dr = p(x^2+y^2) * ds + q(2xy) * ds
```

where 's' is an auxiliary variable.

Solving this system of equations can be challenging and may require specific techniques to find a complete integral.

### Conclusion

Solving non-linear first-order PDEs like (x^2+y^2)p+2xyq=(x+y)z can be complex. Lagrange's method and Charpit's method provide frameworks to approach the solution, although finding a complete integral might require advanced techniques. Remember, the solution will usually involve an arbitrary function, leading to a family of solutions.