## Solving the Differential Equation: (x^4y^4+x^2y^2+xy)ydx+(x^4y^4-x^2y^2+xy)xdy=0

This article explores the solution process for the given differential equation:

**(x^4y^4+x^2y^2+xy)ydx+(x^4y^4-x^2y^2+xy)xdy=0**

This equation is a **first-order homogeneous differential equation**. We can solve it using the following steps:

### 1. Rearranging the Equation

First, let's rearrange the equation to make it easier to work with. Divide both sides by **xy**:

**(x^3y^3 + xy + 1)ydx + (x^3y^3 - xy + 1)xdy = 0**

Now, divide both sides by **(x^3y^3 + xy + 1)(x^3y^3 - xy + 1)**:

**(ydx)/(x^3y^3 - xy + 1) + (xdy)/(x^3y^3 + xy + 1) = 0**

### 2. Substitution and Integration

Let's make the following substitution:

**u = xy**

Then, **du = xdy + ydx**. Substituting these into our rearranged equation gives:

**(du)/(u^3 - u + 1) = 0**

This equation is now separable. Integrating both sides, we get:

**∫(du)/(u^3 - u + 1) = ∫0 du**

The integral on the left-hand side can be solved using partial fractions. However, the exact solution involves complex numbers and is quite intricate.

### 3. General Solution

After solving the integral on the left-hand side, we'll obtain a function of **u** equal to a constant. Substituting back **u = xy** gives us the **general solution** of the differential equation in terms of **x** and **y**.

### Conclusion

While the complete solution involves complex integrals, the process outlined above provides a general framework for solving the given differential equation. The key steps involve rearranging the equation, making a suitable substitution, and integrating both sides. This method can be applied to similar homogeneous differential equations.