## Solving the Differential Equation: (x^2 + y^2 - 5)dx - (y + xy)dy = 0

This article will explore the solution to the differential equation:

**(x^2 + y^2 - 5)dx - (y + xy)dy = 0**

This is a first-order, non-linear differential equation. We can solve this equation by utilizing the method of **exact differential equations**.

### 1. Identifying an Exact Differential Equation

A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is considered exact if:

∂M/∂y = ∂N/∂x

In our case, M(x, y) = x^2 + y^2 - 5 and N(x, y) = -(y + xy). Let's calculate the partial derivatives:

∂M/∂y = 2y ∂N/∂x = -y

Since ∂M/∂y ≠ ∂N/∂x, the given equation is **not exact**.

### 2. Finding an Integrating Factor

We can convert the given equation into an exact one by multiplying it with an **integrating factor**.

To find the integrating factor, we can utilize the following formula:

μ(x) = exp(∫(∂N/∂x - ∂M/∂y)/M dx)

In our case:

μ(x) = exp(∫(-y - 2y)/(x^2 + y^2 - 5) dx) = exp(∫(-3y)/(x^2 + y^2 - 5) dx)

The integral in the exponent does not have a simple solution. However, if we observe that the expression in the denominator (x^2 + y^2 - 5) is similar to the expression in M(x, y), we can try a different approach. Let's assume the integrating factor is of the form μ(y) instead of μ(x).

μ(y) = exp(∫(∂M/∂y - ∂N/∂x)/N dy) = exp(∫(2y + y)/(y + xy) dy) = exp(∫(3y)/(y + xy) dy) = exp(∫(3)/(1 + x) dy) = **(1 + x)^3**

Now, multiplying the original equation with this integrating factor:

(1 + x)^3 * ((x^2 + y^2 - 5)dx - (y + xy)dy) = 0

This equation is now exact since:

∂[(1 + x)^3 * (x^2 + y^2 - 5)]/∂y = 2y(1 + x)^3 ∂[-(1 + x)^3 * (y + xy)]/∂x = 2y(1 + x)^3

### 3. Solving the Exact Equation

Since the equation is exact, we can find a solution by integrating M(x, y) with respect to x and N(x, y) with respect to y.

∫[(1 + x)^3 * (x^2 + y^2 - 5)] dx = ∫[-(1 + x)^3 * (y + xy)] dy

After integrating, we get:

(1/4)(1 + x)^4 + (1/2)y^2(1 + x)^3 - 5(1 + x)^3 = - (1/2)y^2(1 + x)^3 + C

Simplifying and rearranging, we get the final solution:

**(1/4)(1 + x)^4 + y^2(1 + x)^3 - 5(1 + x)^3 = C**

Where C is the constant of integration.

### Conclusion

We successfully solved the non-linear differential equation (x^2 + y^2 - 5)dx - (y + xy)dy = 0 by utilizing the method of exact differential equations. We found that the equation was not exact initially but became exact after multiplying it with the integrating factor (1 + x)^3. The final solution to the equation is **(1/4)(1 + x)^4 + y^2(1 + x)^3 - 5(1 + x)^3 = C**. This solution represents a family of curves in the xy-plane.