## Solving the Differential Equation: (x^2+y^2+1)dx + x(x-2y)dy = 0

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

**(x^2+y^2+1)dx + x(x-2y)dy = 0**

This equation is a **non-exact first-order differential equation**, meaning it cannot be directly solved by integrating both sides. However, we can use an **integrating factor** to make it exact.

### 1. Identifying the Integrating Factor

To find the integrating factor, we need to check if the equation satisfies the following condition:

**(∂M/∂y) = (∂N/∂x)**

Where:

**M = (x^2+y^2+1)****N = x(x-2y)**

Let's calculate the partial derivatives:

**∂M/∂y = 2y****∂N/∂x = 2x - 2y**

Since (∂M/∂y) ≠ (∂N/∂x), the equation is not exact.

Now, let's find the integrating factor **μ(x,y)**:

**μ(x,y) = e^(∫[(∂N/∂x) - (∂M/∂y)]/M dx)**

Plugging in the values:

**μ(x,y) = e^(∫[(2x - 2y) - (2y)]/(x^2+y^2+1) dx)****μ(x,y) = e^(∫(2x - 4y)/(x^2+y^2+1) dx)**

We can simplify this integral by splitting it into two parts:

**μ(x,y) = e^(∫(2x)/(x^2+y^2+1) dx - ∫(4y)/(x^2+y^2+1) dx)**

Solving the first integral, we get:

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

The second integral cannot be solved directly. However, since it only involves 'y', we can treat it as a constant for this integration.

Therefore, the integrating factor is:

**μ(x,y) = e^(ln(x^2+y^2+1) - (4y/x^2+y^2+1))****μ(x,y) = (x^2+y^2+1)e^(-4y/(x^2+y^2+1))**

### 2. Making the Equation Exact

Now, we multiply the original equation by the integrating factor:

**(x^2+y^2+1)e^(-4y/(x^2+y^2+1)) [(x^2+y^2+1)dx + x(x-2y)dy] = 0**

Simplifying, we get:

**(x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1)) dx + x(x-2y)(x^2+y^2+1)e^(-4y/(x^2+y^2+1)) dy = 0**

Now, the equation is exact because:

**∂/∂y [(x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1))] = ∂/∂x [x(x-2y)(x^2+y^2+1)e^(-4y/(x^2+y^2+1))]**

### 3. Solving the Exact Equation

Since the equation is exact, it can be written as the total differential of a function **u(x,y)**:

**du = (∂u/∂x)dx + (∂u/∂y)dy**

Comparing this to our exact equation, we get:

**(∂u/∂x) = (x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1))****(∂u/∂y) = x(x-2y)(x^2+y^2+1)e^(-4y/(x^2+y^2+1))**

Integrating the first equation with respect to 'x', we get:

**u(x,y) = ∫(x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1)) dx + h(y)**

Where h(y) is an arbitrary function of 'y'.

To find h(y), we differentiate this expression with respect to 'y' and compare it to the second equation:

**∂u/∂y = ∂/∂y [∫(x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1)) dx + h(y)]**

After performing the differentiation and comparing with the second equation, we find:

**h'(y) = 0**

Therefore, h(y) is a constant.

### 4. The General Solution

Finally, the general solution to the given differential equation is:

**u(x,y) = ∫(x^2+y^2+1)^2 e^(-4y/(x^2+y^2+1)) dx + C**

Where C is an arbitrary constant.

This integral is complex and cannot be solved analytically. However, we have successfully reduced the original non-exact equation to a form where the solution can be expressed in terms of an integral.

**Note:** This solution might require numerical methods or specific techniques depending on the context and desired accuracy.