The solution of `(dy)/(dx)+2y tanx=sinx,` is

Options

A. `y sec^(3)x=sec^(2)x+C`
B. `y sec^(2)x=sec x+C`
C. `y sin x=tanx+C`
D. none of these

Answer

To solve the differential equation

\[ \frac{dy}{dx} + 2y \tan x = \sin x, \]

we will follow these steps:

### Step 1: Identify \( p \) and \( Q \)

The given equation is in the standard form

\[ \frac{dy}{dx} + p y = Q, \]

where \( p = 2 \tan x \) and \( Q = \sin x \).

### Step 2: Find the Integrating Factor

The integrating factor \( I.F. \) is given by

\[ I.F. = e^{\int p \, dx} = e^{\int 2 \tan x \, dx}. \]

To compute the integral:

\[ \int 2 \tan x \, dx = 2 \ln |\sec x| + C = \ln |\sec^2 x| + C. \]

Thus, the integrating factor is:

\[ I.F. = e^{\ln |\sec^2 x|} = \sec^2 x. \]

### Step 3: Multiply the Differential Equation by the Integrating Factor

Now we multiply the entire differential equation by the integrating factor:

\[ \sec^2 x \frac{dy}{dx} + 2y \sec^2 x \tan x = \sec^2 x \sin x. \]

### Step 4: Rewrite the Left Side

The left-hand side can be rewritten as the derivative of a product:

\[ \frac{d}{dx}(y \sec^2 x) = \sec^2 x \sin x. \]

### Step 5: Integrate Both Sides

Now we integrate both sides:

\[ \int \frac{d}{dx}(y \sec^2 x) \, dx = \int \sec^2 x \sin x \, dx. \]

The left side simplifies to:

\[ y \sec^2 x = \int \sec^2 x \sin x \, dx. \]

### Step 6: Solve the Right Side Integral

To solve the integral on the right, we can use integration by parts or recognize that:

\[ \int \sec^2 x \sin x \, dx = \int \tan x \sec x \, dx. \]

Using the known integral:

\[ \int \sec x \tan x \, dx = \sec x + C. \]

Thus,

\[ y \sec^2 x = \sec x + C. \]

### Step 7: Solve for \( y \)

Now we can isolate \( y \):

\[ y = \sec x + C \cos^2 x. \]

### Final Solution

The solution of the differential equation is:

\[ y = \sec x + C \cos^2 x. \]