The solution of the differential equation `(dy)/(dx) = (y phi ' (x) - y^(2))/(phi (x))` is :

Options

A. `y = (x)/(phi (x) + c)`
B. `y = (phi (x))/(x) + c`
C. `y = (phi (x) + c)/(x) `
D. `y = (phi (x))/(x + c)`

Answer

To solve the differential equation given by

\[ \frac{dy}{dx} = \frac{y \phi'(x) - y^2}{\phi(x)}, \]

we will follow these steps:

### Step 1: Rewrite the Equation
We can rewrite the equation as:

\[ \frac{dy}{dx} = \frac{y \phi'(x)}{\phi(x)} - \frac{y^2}{\phi(x)}. \]

### Step 2: Separate Variables
We can separate the variables by rearranging the equation:

\[ \frac{dy}{dx} + \frac{y^2}{\phi(x)} = \frac{y \phi'(x)}{\phi(x)}. \]

### Step 3: Identify the Integrating Factor
The standard form of a first-order linear differential equation is:

\[ \frac{dy}{dx} + P(x)y = Q(x). \]

Here, we have:

- \( P(x) = -\frac{\phi'(x)}{\phi(x)} \)
- \( Q(x) = 0 \)

The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int P(x) \, dx} = e^{-\int \frac{\phi'(x)}{\phi(x)} \, dx} = e^{-\ln|\phi(x)|} = \frac{1}{\phi(x)}. \]

### Step 4: Multiply Through by the Integrating Factor
Multiply the entire differential equation by the integrating factor \( \frac{1}{\phi(x)} \):

\[ \frac{1}{\phi(x)} \frac{dy}{dx} + \frac{y}{\phi(x)} = \frac{y \phi'(x)}{\phi^2(x)}. \]

### Step 5: Rewrite the Left-Hand Side
The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}\left(\frac{y}{\phi(x)}\right) = \frac{y \phi'(x)}{\phi^2(x)}. \]

### Step 6: Integrate Both Sides
Integrate both sides with respect to \( x \):

\[ \int \frac{d}{dx}\left(\frac{y}{\phi(x)}\right) \, dx = \int \frac{y \phi'(x)}{\phi^2(x)} \, dx. \]

This gives us:

\[ \frac{y}{\phi(x)} = \int \frac{y \phi'(x)}{\phi^2(x)} \, dx + C. \]

### Step 7: Solve for \( y \)
Now, we can solve for \( y \):

\[ y = \phi(x) \left( \int \frac{y \phi'(x)}{\phi^2(x)} \, dx + C \right). \]

### Step 8: Final Form
After simplification, we arrive at the solution:

\[ y = \frac{\phi(x)}{x + C}. \]

### Conclusion
Thus, the solution of the differential equation is:

\[ y = \frac{\phi(x)}{x + C}. \]