COLVE THE FOLOMING $$ \frac{d x}{z(x+y)}=\frac{d y}{z(x-y)}=\frac{d z}{x^{2}+y^{2}} $$

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To solve the system of equations given by

$$
\frac{dx}{z(x+y)} = \frac{dy}{z(x-y)} = \frac{dz}{x^2 + y^2},
$$

we can introduce a parameter $$ t $$ such that:

$$
\frac{dx}{z(x+y)} = \frac{dy}{z(x-y)} = \frac{dz}{x^2 + y^2} = dt.
$$

This gives us three separate equations:

1. $$ \frac{dx}{dt} = z(x+y) $$
2. $$ \frac{dy}{dt} = z(x-y) $$
3. $$ \frac{dz}{dt} = x^2 + y^2 $$

Now, we can analyze these equations step by step.

### Step 1: Solve for $$ z $$

From the first two equations, we can express $$ z $$ in terms of $$ x $$ and $$ y $$:

From equation 1:
$$
z = \frac{dx/dt}{x+y}
$$

From equation 2:
$$
z = \frac{dy/dt}{x-y}
$$

Setting these equal gives us:
$$
\frac{dx/dt}{x+y} = \frac{dy/dt}{x-y}.
$$

Cross-multiplying yields:
$$
(dx/dt)(x-y) = (dy/dt)(x+y).
$$

### Step 2: Solve the system of equations

This equation can be rearranged to:
$$
dx/dt \cdot (x - y) - dy/dt \cdot (x + y) = 0.
$$

This suggests a relationship between $$ x $$ and $$ y $$. We can also express $$ dy/dx $$ in terms of $$ x $$ and $$ y $$:

$$
\frac{dy}{dx} = \frac{z(x-y)}{z(x+y)} = \frac{x-y}{x+y}.
$$

This is a separable differential equation. We can separate variables:

$$
\frac{dy}{x-y} = \frac{dx}{x+y}.
$$

Integrating both sides will help us find a relationship between $$ x $$ and $$ y $$.

### Step 3: Integrate

Integrating the left side:
$$
\int \frac{dy}{x-y} = -\ln|x-y| + C_1,
$$

Integrating the right side:
$$
\int \frac{dx}{x+y} = \ln|x+y| + C_2.
$$

Setting these equal gives us:
$$
-\ln|x-y| = \ln|x+y| + C,
$$
where $$ C = C_2 - C_1 $$.

Exponentiating both sides leads to:
$$
\frac{1}{|x-y|} = K |x+y|,
$$
where $$ K = e^{-C} $$.

### Step 4: Solve for $$ z $$

Now we can substitute back to find $$ z $$. Using the expression for $$ z $$ from either of the original equations, we can find $$ z $$ in terms of $$ x $$ and $$ y $$.

### Conclusion

The solution involves finding the relationships between $$ x $$, $$ y $$, and $$ z $$ through integration and substitution. The final solution will depend on the initial conditions provided.

If you have specific initial conditions or further constraints, please provide them for a more tailored solution.
The system of equations can be solved by introducing a parameter $$ t $$ and expressing $$ z $$ in terms of $$ x $$ and $$ y $$. Then, a relationship between $$ x $$ and $$ y $$ is found through integration, and $$ z $$ is determined using the expressions for $$ z $$ from the original equations. The final solution depends on the initial conditions.

COLVE THE FOLOMING \( \frac{d x}{z(x+y)}=\frac{d y}{z(x-y)}=\frac{d z}{x^{2}+y^{2}} \)

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