The solution of `(dy)/(dx) + y tan x = sec x ` is

Options

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

Explanation

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

### Step 1: Identify \(p(x)\) and \(q(x)\)
The given equation is in the standard form of a linear differential equation:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
Here, we have:
- \(p(x) = \tan x\)
- \(q(x) = \sec x\)

### Step 2: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int p(x) \, dx} = e^{\int \tan x \, dx} \]
We know that:
\[ \int \tan x \, dx = -\ln(\cos x) = \ln(\sec x) \]
Thus, the integrating factor becomes:
\[ I(x) = e^{\ln(\sec x)} = \sec x \]

### Step 3: Multiply the Entire Equation by the Integrating Factor
Now we multiply the entire differential equation by the integrating factor \(\sec x\):
\[ \sec x \frac{dy}{dx} + y \sec x \tan x = \sec^2 x \]

### Step 4: Rewrite the Left Side as a Derivative
The left-hand side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(y \sec x) = \sec^2 x \]

### Step 5: Integrate Both Sides
Now we integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(y \sec x) \, dx = \int \sec^2 x \, dx \]
This gives us:
\[ y \sec x = \tan x + C \]
where \(C\) is the constant of integration.

### Step 6: Solve for \(y\)
Now, we solve for \(y\):
\[ y = \tan x \cos x + C \cos x \]
Since \(\tan x = \frac{\sin x}{\cos x}\), we can simplify:
\[ y = \sin x + C \cos x \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = \sin x + C \cos x \]

---