If `phi(x)` is a differentiable function, then the solution of the different equation `dy+{y phi' (x) - phi (x) phi'(x)} dx=0,` is

Options

A. `y={phi(x)-1}+Ce^(-phi(x))`
B. `yphi(x)={phi(x)}^(2)+C`
C. `ye^(phi(x))=phi(x)e^(phi(x))+C`
D. `y-phi(x)=phi(x)e^(-phi(x))`

Explanation

To solve the differential equation \( dy + (y \phi'(x) - \phi(x) \phi'(x)) dx = 0 \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ dy + (y \phi'(x) - \phi(x) \phi'(x)) dx = 0 \]
Dividing both sides by \( dx \), we get:
\[ \frac{dy}{dx} + y \phi'(x) - \phi(x) \phi'(x) = 0 \]

### Step 2: Rearranging the Equation
Rearranging the equation gives us:
\[ \frac{dy}{dx} + y \phi'(x) = \phi(x) \phi'(x) \]
This is now in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = \phi'(x) \) and \( Q(x) = \phi(x) \phi'(x) \).

### Step 3: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is calculated as:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \phi'(x) \, dx} = e^{\phi(x)} \]

### Step 4: Multiply the Equation by the Integrating Factor
Now we multiply the entire equation by the integrating factor:
\[ e^{\phi(x)} \frac{dy}{dx} + e^{\phi(x)} y \phi'(x) = e^{\phi(x)} \phi(x) \phi'(x) \]

### Step 5: Recognizing the Left Side as a Derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dx}(y e^{\phi(x)}) = e^{\phi(x)} \phi(x) \phi'(x) \]

### Step 6: Integrate Both Sides
Integrating both sides with respect to \( x \):
\[ y e^{\phi(x)} = \int e^{\phi(x)} \phi(x) \phi'(x) \, dx + C \]
Let \( t = \phi(x) \), then \( dt = \phi'(x) dx \), and the integral becomes:
\[ \int t e^t \, dt \]

### Step 7: Solve the Integral Using Integration by Parts
Using integration by parts, let:
- \( u = t \) and \( dv = e^t dt \)
Then:
- \( du = dt \) and \( v = e^t \)

Thus, we have:
\[ \int t e^t \, dt = t e^t - \int e^t \, dt = t e^t - e^t + C \]
Substituting back, we get:
\[ y e^{\phi(x)} = \phi(x) e^{\phi(x)} - e^{\phi(x)} + C \]

### Step 8: Solve for \( y \)
Dividing by \( e^{\phi(x)} \):
\[ y = \phi(x) - 1 + C e^{-\phi(x)} \]

### Final Solution
Thus, the solution of the differential equation is:
\[ y = \phi(x) - 1 + C e^{-\phi(x)} \]