Find the general solution of the differential equation
`y-x(dy)/(dx)=a(y^(2)+(dy)/(dx))`.

Options

A. `y=C(1-ay)(a+x)`
B. `y=C(1+ay)(a+x)`
C. `y=C(a+x)`
D. `y=C(1-ay)`

Explanation

To solve the differential equation \( y - x \frac{dy}{dx} = a(y^2 + \frac{dy}{dx}) \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the terms to isolate \( \frac{dy}{dx} \):
\[ y - ay^2 = a + x \frac{dy}{dx} \]
This can be rewritten as:
\[ y(1 - ay) = a + x \frac{dy}{dx} \]

### Step 2: Isolate \( \frac{dy}{dx} \)
Next, we isolate \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} = y(1 - ay) - a \]
Thus,
\[ \frac{dy}{dx} = \frac{y(1 - ay) - a}{x} \]

### Step 3: Separation of Variables
Now we separate the variables:
\[ \frac{dy}{y(1 - ay) - a} = \frac{dx}{x} \]

### Step 4: Integrating Both Sides
We will integrate both sides. The left side requires partial fraction decomposition:
\[ \frac{1}{y(1 - ay) - a} = \frac{A}{y} + \frac{B}{1 - ay} \]
Finding \( A \) and \( B \) gives:
\[ 1 = A(1 - ay) + By \]
Setting up equations for coefficients, we can solve for \( A \) and \( B \).

After finding \( A \) and \( B \), we integrate:
\[ \int \left( \frac{A}{y} + \frac{B}{1 - ay} \right) dy = \int \frac{dx}{x} \]

### Step 5: Solving the Integrals
The integrals yield:
\[ A \ln |y| - B \ln |1 - ay| = \ln |x| + C \]
Where \( C \) is the constant of integration.

### Step 6: Exponentiating Both Sides
Exponentiating both sides gives:
\[ \frac{y^A}{(1 - ay)^B} = kx \]
Where \( k = e^C \).

### Step 7: Final Solution
Rearranging this gives us the general solution in terms of \( y \):
\[ y = \frac{kx(1 - ay)^B}{y^A} \]

### Summary
The general solution of the differential equation is:
\[ y = C(1 - ay)x \]
where \( C \) is a constant.