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 will follow a systematic approach.

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

### Step 2: Substitute \( \sin y \) with \( t \)
Let \( t = \sin y \). Then, differentiating both sides with respect to \( x \) gives:
\[ \frac{dt}{dx} = \cos y \frac{dy}{dx} \]
Substituting this into our equation, we have:
\[ x \frac{dt}{dx} = 1 - t \]

### Step 3: Rearranging the Equation
Now we can rewrite the equation as:
\[ \frac{dt}{dx} + \frac{t}{x} = \frac{1}{x} \]
This is a linear first-order differential equation in \( t \).

### Step 4: Identify the Integrating Factor
The standard form of a linear differential equation is:
\[ \frac{dt}{dx} + p(x)t = q(x) \]
where \( p(x) = \frac{1}{x} \) and \( q(x) = \frac{1}{x} \). The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln x} = x \]

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

### Step 6: Integrate Both Sides
Now, we can integrate both sides:
\[ \int \left( x \frac{dt}{dx} + t \right) dx = \int 1 \, dx \]
This simplifies to:
\[ tx = x + \lambda \]
where \( \lambda \) is an arbitrary constant.

### Step 7: Substitute Back for \( t \)
Since we set \( t = \sin y \), we substitute back:
\[ x \sin y = x + \lambda \]

### Step 8: Rearranging the Final Equation
Rearranging gives us:
\[ x - x \sin y = \lambda \]
or
\[ x(1 - \sin y) = \lambda \]

### Final Solution
Thus, the solution of the differential equation is:
\[ x - x \sin y = \lambda \]