The solution of the differential equation ` (dy)/(dx) = sec x -y tan x ` is

Options

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

Answer

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

### Step 1: Rewrite the equation
We can rearrange the equation into the standard form of a linear differential equation:
\[ \frac{dy}{dx} + y \tan x = \sec x \]

### Step 2: Identify \(p(x)\) and \(q(x)\)
In the standard form \(\frac{dy}{dx} + p(x)y = q(x)\), we have:
- \(p(x) = \tan x\)
- \(q(x) = \sec x\)

### Step 3: 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:
\[ \int \tan x \, dx = -\log(\cos x) = \log(\sec x) \]
Thus, the integrating factor becomes:
\[ I(x) = e^{\log(\sec x)} = \sec x \]

### Step 4: Multiply through 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 5: Recognize the left-hand side as a derivative
The left-hand side can be recognized as the derivative of the product \(y \sec x\):
\[ \frac{d}{dx}(y \sec x) = \sec^2 x \]

### Step 6: 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 7: 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 it as:
\[ y = \sin x + C \cos x \]

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

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