The solution of the differential equation `sin 2y (dy)/(dx) +2 tan x cos ^(2) y=2 sec x cos ^(3) y` is: (where C is arbitary constant)

Options

A. `cos y sec x= tan x+C`
B. ` sec y cos x = tan x +C`
C. `sec y sec x = tan x +C`
D. `tan y sec x = sec x +C`

Answer

To solve the differential equation \( \sin 2y \frac{dy}{dx} + 2 \tan x \cos^2 y = 2 \sec x \cos^3 y \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ \sin 2y \frac{dy}{dx} + 2 \tan x \cos^2 y = 2 \sec x \cos^3 y \]

### Step 2: Simplify the equation
Divide the entire equation by \( \cos^3 y \):
\[ \frac{\sin 2y}{\cos^3 y} \frac{dy}{dx} + 2 \tan x \frac{\cos^2 y}{\cos^3 y} = 2 \sec x \]
This simplifies to:
\[ \frac{\sin 2y}{\cos^3 y} \frac{dy}{dx} + 2 \tan x \sec y = 2 \sec x \]

### Step 3: Express \(\sin 2y\)
Recall that \( \sin 2y = 2 \sin y \cos y \). Substitute this into the equation:
\[ \frac{2 \sin y \cos y}{\cos^3 y} \frac{dy}{dx} + 2 \tan x \sec y = 2 \sec x \]
This simplifies to:
\[ \frac{2 \sin y}{\cos^2 y} \frac{dy}{dx} + 2 \tan x \sec y = 2 \sec x \]

### Step 4: Isolate \(\frac{dy}{dx}\)
Rearranging gives:
\[ \frac{2 \sin y}{\cos^2 y} \frac{dy}{dx} = 2 \sec x - 2 \tan x \sec y \]
Dividing by 2:
\[ \frac{\sin y}{\cos^2 y} \frac{dy}{dx} = \sec x - \tan x \sec y \]

### Step 5: Let \( t = \sec y \)
Let \( t = \sec y \), then \( \frac{dy}{dx} = \frac{dt}{dx} \cdot \frac{1}{t \tan y} \):
\[ \frac{dt}{dx} \cdot \frac{1}{t \tan y} = \sec x - \tan x t \]
This can be rearranged to:
\[ \frac{dt}{dx} = t \tan y \left( \sec x - \tan x t \right) \]

### Step 6: Find the integrating factor
The equation is now in the form of a linear differential equation. The integrating factor is given by:
\[ e^{\int P(x) dx} \]
where \( P(x) = \tan x \). Thus, the integrating factor is:
\[ e^{\int \tan x \, dx} = e^{-\ln \cos x} = \sec x \]

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

### Step 8: Solve the linear equation
This can be integrated:
\[ \frac{d}{dx}(t \sec x) = \sec^2 x \]
Integrating both sides:
\[ t \sec x = \tan x + C \]
where \( C \) is the constant of integration.

### Step 9: Substitute back for \( t \)
Recall that \( t = \sec y \):
\[ \sec y \sec x = \tan x + C \]

### Final Step: Rearranging the equation
Thus, we have:
\[ \sec y = \frac{\tan x + C}{\sec x} \]

### Conclusion
The solution of the differential equation is:
\[ \sec y \sec x = \tan x + C \]