## Solving the Differential Equation (x^2 + y^2 - 5)dx = (y + xy)dy with Initial Condition y(0) = 1

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

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

with the initial condition **y(0) = 1**.

### 1. Rearranging the Equation

First, we need to rearrange the equation into a more manageable form. Divide both sides by dx and rearrange the terms:

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

### 2. Identifying the Type of Differential Equation

The equation is **non-linear** and **not separable**. However, we can try to solve it using an **integrating factor**. To do this, we need to rewrite the equation in the form:

**dy/dx + P(x)y = Q(x)**

To achieve this, we can factor out y from the right-hand side:

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

Then, we can rewrite it as:

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

Now, we have the equation in the desired form, with **P(x) = -1** and **Q(x) = (x^2 - 5) / (y + xy)**.

### 3. Finding the Integrating Factor

The integrating factor is given by:

**μ(x) = exp(∫P(x) dx)**

In our case:

**μ(x) = exp(∫-1 dx) = exp(-x)**

### 4. Multiplying the Equation by the Integrating Factor

Multiplying both sides of the equation by the integrating factor:

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

The left-hand side now represents the derivative of the product of y and the integrating factor:

**d/dx (y * exp(-x)) = (x^2 - 5) exp(-x) / (y + xy)**

### 5. Integrating Both Sides

Integrate both sides with respect to x:

**∫d/dx (y * exp(-x)) dx = ∫(x^2 - 5) exp(-x) / (y + xy) dx**

This simplifies to:

**y * exp(-x) = ∫(x^2 - 5) exp(-x) / (y + xy) dx + C**

Where C is the constant of integration.

### 6. Solving for y

To solve for y, we need to evaluate the integral on the right-hand side. This integral is complex and may not have a closed-form solution. We can use numerical methods or approximations to find a solution.

### 7. Applying the Initial Condition

Finally, we can use the initial condition **y(0) = 1** to solve for the constant C:

**1 * exp(0) = ∫(0^2 - 5) exp(0) / (1 + 0*1) d0 + C**

This simplifies to:

**1 = -5 + C**

Therefore, **C = 6**.

### 8. Finding the Solution

Substituting C back into the equation and solving for y, we obtain the solution to the differential equation. This will be an implicit solution, as we might not be able to isolate y explicitly.

### Conclusion

Solving the given differential equation requires several steps, including rearranging the equation, identifying the type, finding the integrating factor, and integrating both sides. While the final solution might be complex, understanding the process is key to solving similar problems. Remember that the initial condition is crucial to determine the specific solution.