The general solution of the differential equation `(dy)/(dx) = y tan x - y^(2) sec x` is

Options

A. `tan x = (c + sec x)y`
B. `sec y = (c + tan y)x`
C. `sec x = (c + tan x) y`
D. None of these

Explanation

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

### Step 1: Rewrite the equation
We start with the given equation:
\[ \frac{dy}{dx} = y \tan x - y^2 \sec x \]
We can rearrange it to isolate the terms involving \(y\):
\[ \frac{dy}{dx} + y^2 \sec x = y \tan x \]

### Step 2: Divide by \(y^2\)
Next, we divide the entire equation by \(y^2\):
\[ \frac{1}{y^2} \frac{dy}{dx} + \sec x = \frac{\tan x}{y} \]

### Step 3: Substitute \(t = \frac{1}{y}\)
Let us make the substitution \(t = \frac{1}{y}\). Then, we have:
\[ y = \frac{1}{t} \quad \text{and} \quad \frac{dy}{dx} = -\frac{1}{t^2} \frac{dt}{dx} \]
Substituting these into the equation gives:
\[ -\frac{1}{t^2} \frac{dt}{dx} + \sec x = t \tan x \]
Rearranging this, we have:
\[ -\frac{1}{t^2} \frac{dt}{dx} = t \tan x - \sec x \]

### Step 4: Multiply through by \(-t^2\)
Multiplying through by \(-t^2\) gives:
\[ \frac{dt}{dx} = t^3 \tan x - t^2 \sec x \]

### Step 5: Rearranging the equation
Rearranging the equation, we have:
\[ \frac{dt}{dx} + t^3 \tan x = t^2 \sec x \]

### Step 6: Integrating factor
The integrating factor \(I\) is given by:
\[ I = e^{\int \tan x \, dx} = e^{-\ln(\cos x)} = \sec x \]

### Step 7: Multiply through by the integrating factor
Multiplying the entire equation by \(\sec x\):
\[ \sec x \frac{dt}{dx} + t^3 \sec x \tan x = t^2 \]

### Step 8: Solve the left-hand side
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}(t^2 \sec x) = t^2 \]

### Step 9: Integrate both sides
Integrating both sides:
\[ \int \frac{d}{dx}(t^2 \sec x) \, dx = \int t^2 \, dx \]
This gives:
\[ t^2 \sec x = \tan x + C \]

### Step 10: Substitute back for \(y\)
Substituting back \(t = \frac{1}{y}\):
\[ \frac{1}{y^2} \sec x = \tan x + C \]
Thus,
\[ \sec x = y^2 (\tan x + C) \]

### Final Solution
The general solution of the differential equation is:
\[ \sec x = y^2 (\tan x + C) \]