## Solving the Differential Equation: (xy+x)dx = (x^2y^2 + x^2 + y^2 + 1)dy

This article will guide you through the process of solving the differential equation:

**(xy + x)dx = (x^2y^2 + x^2 + y^2 + 1)dy**

This equation appears complex, but we can simplify it and solve it using techniques from differential equation theory.

### 1. Rearranging the Equation

First, we need to rewrite the equation in a form that is easier to work with. We can achieve this by dividing both sides by dx and rearranging:

**(xy + x) = (x^2y^2 + x^2 + y^2 + 1)dy/dx**

### 2. Recognizing the Form

Observe that the right side of the equation can be factored:

**(xy + x) = [(x^2 + 1)(y^2 + 1)]dy/dx**

This form suggests that we might be dealing with a **separable differential equation**, where we can separate the variables x and y on opposite sides.

### 3. Separating the Variables

To separate the variables, let's divide both sides by (x^2 + 1) and (y^2 + 1):

**(xy + x) / [(x^2 + 1)(y^2 + 1)] = dy/dx**

Now, we can rewrite the left side by splitting the fraction:

**(x/(x^2 + 1)) * (y/(y^2 + 1)) = dy/dx**

Finally, we can separate the variables:

**(y/(y^2 + 1)) dy = (x/(x^2 + 1)) dx**

### 4. Integrating Both Sides

Now we have the variables separated. To solve for y, we integrate both sides:

**∫ (y/(y^2 + 1)) dy = ∫ (x/(x^2 + 1)) dx**

These are standard integrals:

**1/2 * ln(y^2 + 1) = 1/2 * ln(x^2 + 1) + C**

where C is the constant of integration.

### 5. Solving for y

To solve for y explicitly, we can simplify and rearrange:

**ln(y^2 + 1) = ln(x^2 + 1) + 2C**

**y^2 + 1 = e^(ln(x^2 + 1) + 2C)**

**y^2 + 1 = (x^2 + 1) * e^(2C)**

Let's denote e^(2C) as a new constant K:

**y^2 + 1 = K(x^2 + 1)**

Finally, solving for y:

**y^2 = K(x^2 + 1) - 1**

**y = ±√[K(x^2 + 1) - 1]**

### Conclusion

The solution to the differential equation (xy + x)dx = (x^2y^2 + x^2 + y^2 + 1)dy is:

**y = ±√[K(x^2 + 1) - 1]**

where K is an arbitrary constant. This solution represents a family of curves, each determined by a specific value of K.