The Solution of the equation `(dy)/(dx)+y tan x =sec x` is

Options

A. `y=sin x +c cos x`
B. `y= sin x -c cos x`
C. `y=tan x+cot x +c`
D. None of these

Answer

To solve the differential equation \(\frac{dy}{dx} + y \tan x = \sec x\), we will follow the steps for solving a first-order linear differential equation.

### Step 1: Identify \(p(x)\) and \(q(x)\)
The given equation can be rewritten in the standard form:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
where \(p(x) = \tan x\) and \(q(x) = \sec x\).

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

### Step 3: Multiply the entire differential equation by the integrating factor
Multiplying the original equation by \(\sec x\):
\[ \sec x \frac{dy}{dx} + y \sec x \tan x = \sec^2 x \]

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

### Step 5: Integrate both sides
Integrating both sides with respect to \(x\):
\[ \int \frac{d}{dx}(y \sec x) \, dx = \int \sec^2 x \, dx \]
This gives:
\[ y \sec x = \tan x + C \]
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 rewrite:
\[ y = \sin x + C \cos x \]

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