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: Rearranging the Equation
We start by rearranging the given equation:
\[ \frac{dy}{dx} - y \tan x = -y^2 \sec x \]

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

### Step 3: Substituting \(v = \frac{1}{y}\)
Let \(v = \frac{1}{y}\). Then, we differentiate \(y\) with respect to \(x\):
\[ y = \frac{1}{v} \implies \frac{dy}{dx} = -\frac{1}{v^2} \frac{dv}{dx} \]

### Step 4: Substitute into the Equation
Substituting \(\frac{dy}{dx}\) into the rearranged equation gives:
\[ -\frac{1}{v^2} \frac{dv}{dx} - v \tan x = -\sec x \]
Multiplying through by \(-1\):
\[ \frac{1}{v^2} \frac{dv}{dx} + v \tan x = \sec x \]

### Step 5: Rearranging
Rearranging the equation:
\[ \frac{dv}{dx} + v^2 \tan x = v^2 \sec x \]

### Step 6: Identifying as a Linear Differential Equation
This is a first-order linear differential equation in the standard form:
\[ \frac{dv}{dx} + P(x)v = Q(x) \]
where \(P(x) = \tan x\) and \(Q(x) = \sec x\).

### Step 7: Finding the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int P(x) dx} = e^{\int \tan x \, dx} = e^{-\ln(\cos x)} = \sec x \]

### Step 8: Multiplying the Equation by the Integrating Factor
Multiply the entire differential equation by \(\sec x\):
\[ \sec x \frac{dv}{dx} + v \sec x \tan x = 1 \]

### Step 9: Integrating Both Sides
The left side can be written as the derivative of a product:
\[ \frac{d}{dx}(v \sec x) = 1 \]
Integrating both sides:
\[ v \sec x = x + C \]

### Step 10: Substitute Back for \(y\)
Recall that \(v = \frac{1}{y}\):
\[ \frac{1}{y} \sec x = x + C \]
Thus,
\[ \sec x = y(x + C) \]

### Final Solution
Rearranging gives us the general solution of the differential equation:
\[ \sec x = y \tan x + C \]