The solution of the differential equation `x cos y (dy)/(dx)+siny =1` is (Here, `x gt0` and `lambda` is an arbitrary constant)

Options

A. `x-x cos x =lamdba`
B. `x+x cos x = lambda`
C. `x- x sin y = lambda`
D. `x+x cos y= lambda`

Answer

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

### Step 1: Rewrite the equation
We start with the given differential equation:
\[ x \cos y \frac{dy}{dx} + \sin y = 1 \]
We can rearrange this to isolate the term involving \(\frac{dy}{dx}\):
\[ x \cos y \frac{dy}{dx} = 1 - \sin y \]

### Step 2: Substitute \( t = \sin y \)
Let \( t = \sin y \). Then, the derivative of \( t \) with respect to \( x \) is:
\[ \frac{dt}{dx} = \cos y \frac{dy}{dx} \]
Thus, we can express \(\cos y \frac{dy}{dx}\) in terms of \( t \):
\[ \cos y \frac{dy}{dx} = \frac{dt}{dx} \]
Substituting this back into the equation gives:
\[ x \frac{dt}{dx} = 1 - t \]

### Step 3: Divide by \( x \)
Now, we divide the entire equation by \( x \):
\[ \frac{dt}{dx} + \frac{t}{x} = \frac{1}{x} \]

### Step 4: Identify the integrating factor
This is a linear first-order differential equation. The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln x} = x \]

### Step 5: Multiply through by the integrating factor
Multiply the entire equation by the integrating factor \( x \):
\[ x \frac{dt}{dx} + t = 1 \]

### Step 6: Recognize the left-hand side as a derivative
The left-hand side can be recognized as the derivative of a product:
\[ \frac{d}{dx}(xt) = 1 \]

### Step 7: Integrate both sides
Integrating both sides with respect to \( x \):
\[ xt = x + C \]
where \( C \) is the constant of integration.

### Step 8: Solve for \( t \)
Now, we can solve for \( t \):
\[ t = 1 + \frac{C}{x} \]

### Step 9: Substitute back for \( t \)
Recall that \( t = \sin y \):
\[ \sin y = 1 + \frac{C}{x} \]

### Step 10: Rearrange to find the solution
Rearranging gives:
\[ x - x \sin y = C \]
Letting \( C = -\lambda \) (where \( \lambda \) is an arbitrary constant):
\[ x - x \sin y = \lambda \]

Thus, the solution to the differential equation is:
\[ x(1 - \sin y) = \lambda \]