General solution of the differential equation `(cosx)(dy)/(dx)+y.sinx=1` is

Options

A. `y=csinx+cosx`
B. `y=sinx+c cos x`
C. `y=tanx+c`
D. `ysinx=sinx+c`

Answer

To solve the differential equation \((\cos x) \frac{dy}{dx} + y \sin x = 1\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos x \frac{dy}{dx} + y \sin x = 1 \]
To simplify, we divide the entire equation by \(\cos x\):
\[ \frac{dy}{dx} + y \frac{\sin x}{\cos x} = \frac{1}{\cos x} \]
This can be rewritten as:
\[ \frac{dy}{dx} + y \tan x = \sec x \]

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

### Step 3: Find the integrating factor
The integrating factor \(\mu(x)\) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \tan x \, dx} \]
The integral of \(\tan x\) is:
\[ \int \tan x \, dx = -\ln |\cos x| + C \]
Thus, the integrating factor becomes:
\[ \mu(x) = e^{-\ln |\cos x|} = \frac{1}{\cos x} = \sec x \]

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

### Step 6: Integrate both sides
Now we integrate both sides:
\[ \int \frac{d}{dx}(y \sec x) \, dx = \int \sec^2 x \, dx \]
This gives:
\[ y \sec x = \tan x + C \]
where \(C\) is the constant of integration.

### Step 7: Solve for \(y\)
Finally, 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 Answer
Thus, the general solution of the differential equation is:
\[ y = \sin x + C \cos x \]

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