Find the general solution of the following :
`(x+2)(dy)/(dx)=x^(2)+4x-9`.

Answer

To find the general solution of the differential equation

\[ (x+2) \frac{dy}{dx} = x^2 + 4x - 9, \]

we will follow these steps:

### Step 1: Rewrite the equation
First, we can rewrite the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{x^2 + 4x - 9}{x + 2}. \]

### Step 2: Simplify the right-hand side
Next, we will simplify the right-hand side. We can factor the quadratic expression in the numerator:

\[ x^2 + 4x - 9 = (x^2 + 4x + 4) - 13 = (x + 2)^2 - 13. \]

Thus, we can rewrite the equation as:

\[ \frac{dy}{dx} = \frac{(x + 2)^2 - 13}{x + 2}. \]

### Step 3: Split the fraction
Now we can split the fraction:

\[ \frac{dy}{dx} = \frac{(x + 2)^2}{x + 2} - \frac{13}{x + 2} = (x + 2) - \frac{13}{x + 2}. \]

### Step 4: Integrate both sides
Now we will integrate both sides with respect to \(x\):

\[ dy = \left((x + 2) - \frac{13}{x + 2}\right) dx. \]

Integrating both sides:

\[ y = \int \left((x + 2) - \frac{13}{x + 2}\right) dx. \]

### Step 5: Perform the integration
We can integrate each term separately:

1. \(\int (x + 2) \, dx = \frac{x^2}{2} + 2x\)
2. \(\int \frac{13}{x + 2} \, dx = 13 \ln|x + 2|\)

Putting it all together, we have:

\[ y = \frac{x^2}{2} + 2x - 13 \ln|x + 2| + C, \]

where \(C\) is the constant of integration.

### Final Solution
Thus, the general solution of the differential equation is:

\[ y = \frac{x^2}{2} + 2x - 13 \ln|x + 2| + C. \]