General solution of differential equation of ` f(x) (dy)/(dx) = f^(2) (x ) + f(x) y + f'(x) y` is : ( c being arbitrary constant ) .

Options

A. `y= f(x) + ce^(x) `
B. `y=-f(x) + ce^(x) `
C. `y=-f(x) + ce^(x) f(x)`
D. `y=cf(x) + e^(x) `

Answer

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

### Step 1: Rewrite the Equation
Start with the given equation:
\[ f(x) \frac{dy}{dx} = f^2(x) + f(x) y + f'(x) y \]

### Step 2: Rearrange the Terms
Rearranging the equation gives:
\[ f(x) \frac{dy}{dx} - f'(x) y = f^2(x) + f(x) y \]

### Step 3: Factor Out Terms
We can factor out \( y \) from the right-hand side:
\[ f(x) \frac{dy}{dx} - f'(x) y = f^2(x) + f(x) y \]
This can be rewritten as:
\[ f(x) \frac{dy}{dx} - y \frac{d}{dx} f(x) = f^2(x) \]

### Step 4: Multiply by \( \frac{1}{f^2(x)} \)
Now, we will divide the entire equation by \( f^2(x) \):
\[ \frac{1}{f^2(x)} \left( f(x) \frac{dy}{dx} - y \frac{d}{dx} f(x) \right) = 1 \]

### Step 5: Use the Property of Derivatives
Using the property \( \frac{d}{dx} \left( \frac{y}{f(x)} \right) = \frac{f(x) \frac{dy}{dx} - y f'(x)}{f^2(x)} \), we can rewrite the left-hand side:
\[ \frac{d}{dx} \left( \frac{y}{f(x)} \right) = 1 \]

### Step 6: Integrate Both Sides
Integrating both sides gives:
\[ \int \frac{d}{dx} \left( \frac{y}{f(x)} \right) dx = \int 1 \, dx \]
This results in:
\[ \frac{y}{f(x)} = x + C \]
where \( C \) is the constant of integration.

### Step 7: Solve for \( y \)
Multiplying both sides by \( f(x) \) yields:
\[ y = f(x)(x + C) \]

### Final Solution
Thus, the general solution of the differential equation is:
\[ y = f(x)(x + C) \]