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

Options

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

Answer

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

### Step 1: Rearranging the Equation
We start by rearranging the given equation to isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{f^2(x) + yf(x) + f'(x)y}{f(x)} \]

### Step 2: Simplifying the Right Side
We can simplify the right side:
\[ \frac{dy}{dx} = \frac{f^2(x)}{f(x)} + \frac{yf(x)}{f(x)} + \frac{f'(x)y}{f(x)} = f(x) + y + \frac{f'(x)}{f(x)}y \]

### Step 3: Factoring the Equation
Now, we can factor out \(y\) from the terms involving \(y\):
\[ \frac{dy}{dx} = f(x) + y\left(1 + \frac{f'(x)}{f(x)}\right) \]

### Step 4: Introducing a New Variable
Let \(v = y\) and rewrite the equation:
\[ \frac{dv}{dx} = f(x) + v\left(1 + \frac{f'(x)}{f(x)}\right) \]

### Step 5: Separating Variables
We can separate the variables:
\[ \frac{dv}{f(x) + v\left(1 + \frac{f'(x)}{f(x)}\right)} = dx \]

### Step 6: Integrating Both Sides
Now, we integrate both sides. The left side can be integrated using the substitution method or partial fractions if necessary. Let's denote the integral on the left side as \(I\):
\[ I = \int \frac{dv}{f(x) + v\left(1 + \frac{f'(x)}{f(x)}\right)} = \int dx \]

### Step 7: Solving the Integral
After integrating, we will have:
\[ \text{Let } u = f(x) + v\left(1 + \frac{f'(x)}{f(x)}\right) \]
This will give us a logarithmic form after integration.

### Step 8: Back Substituting
After integration, we will back substitute \(v = y\) and express the solution in terms of \(y\):
\[ y = \frac{C e^x - f(x)}{1 + \frac{f'(x)}{f(x)}} \]
where \(C\) is the constant of integration.

### Final Solution
Thus, the general solution of the differential equation is:
\[ y = C e^x - f(x) \]
where \(C\) is an arbitrary constant.

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