## Solving the Differential Equation (x-y-2)dx - (2x-2y-3)dy = 0

This article will walk you through the process of solving the first-order differential equation:

**(x-y-2)dx - (2x-2y-3)dy = 0**

### 1. Recognizing the Type of Differential Equation

The given equation is a **non-exact differential equation**. This means that it doesn't satisfy the condition for exactness:

**∂M/∂y = ∂N/∂x**

Where:

- M(x,y) = x-y-2
- N(x,y) = -(2x-2y-3)

In our case, ∂M/∂y = -1 and ∂N/∂x = -2, which are not equal.

### 2. Finding an Integrating Factor

To solve this non-exact differential equation, we need to find an integrating factor, which will make the equation exact.

**a) Finding the Integrating Factor (IF):**

We can use the following formula to find the integrating factor:

**IF = exp(∫(∂N/∂x - ∂M/∂y) / M dx)**

In our case, this becomes:

**IF = exp(∫((-2) - (-1)) / (x-y-2) dx)****IF = exp(∫(-1) / (x-y-2) dx)****IF = exp(-ln|x-y-2|)****IF = 1/(x-y-2)**

**b) Multiplying the Equation by the Integrating Factor:**

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

**(1/(x-y-2)) * [(x-y-2)dx - (2x-2y-3)dy] = 0****dx - [(2x-2y-3)/(x-y-2)]dy = 0**

This equation is now exact.

### 3. Solving the Exact Differential Equation

Let's rewrite the equation as:

**M(x,y)dx + N(x,y)dy = 0**

Where:

**M(x,y) = 1****N(x,y) = -(2x-2y-3)/(x-y-2)**

Now, we need to find a function u(x,y) such that:

**∂u/∂x = M(x,y)****∂u/∂y = N(x,y)**

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

**u(x,y) = x + g(y)**

Where g(y) is an arbitrary function of y.

Differentiating this expression with respect to y and equating it to N(x,y), we get:

**∂u/∂y = g'(y) = -(2x-2y-3)/(x-y-2)**

This equation doesn't contain x, so we can solve it for g'(y) by setting x = 0:

**g'(y) = (2y+3)/(y+2)**

Integrating g'(y) with respect to y:

**g(y) = 2y + ln|y+2| + C**

Finally, substituting this expression for g(y) back into the equation for u(x,y), we get the solution:

**u(x,y) = x + 2y + ln|y+2| + C = 0**

### 4. Implicit Solution

The solution to the original differential equation is given implicitly by the equation:

**x + 2y + ln|y+2| + C = 0**

Where C is an arbitrary constant.

### Conclusion

We successfully solved the non-exact differential equation (x-y-2)dx-(2x-2y-3)dy=0 by finding an integrating factor, making the equation exact, and then integrating to obtain the implicit solution.