If ` y = y ( x ) ` is the solution of differential equation ` sin y (dy ) /(dx ) - cos y = e ^ ( - x ) ` such that ` y ( 0 ) = ( pi ) /(2) ` then ` y (A) ` is equal to

Options

A. ` sin ^( -1 ) "" (1 ) /(e ) `
B. ` cos ^( -1) "" ( 1 )/(e ) `
C. ` - cos ^( -1) "" (1 ) / (e ) `
D. ` cos ^( -1) ( - (1 ) /(e ) ) `

Explanation

To solve the differential equation \( \sin y \frac{dy}{dx} - \cos y = e^{-x} \) with the initial condition \( y(0) = \frac{\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the equation:
\[ \sin y \frac{dy}{dx} - \cos y = e^{-x} \]
Rearranging gives:
\[ \sin y \frac{dy}{dx} = e^{-x} + \cos y \]

### Step 2: Substitute for \( t \)
Let \( t = -\cos y \). Then, we can differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \sin y \frac{dy}{dx} \]
Thus, we can rewrite the equation as:
\[ \frac{dt}{dx} = e^{-x} + t \]

### Step 3: Formulate the Linear Differential Equation
The equation now is in the form:
\[ \frac{dt}{dx} - t = e^{-x} \]
This is a first-order linear differential equation.

### Step 4: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int -1 \, dx} = e^{-x} \]

### Step 5: Multiply the Equation by the Integrating Factor
Multiplying the entire equation by the integrating factor:
\[ e^{-x} \frac{dt}{dx} - e^{-x} t = 1 \]

### Step 6: Integrate Both Sides
Now we integrate both sides:
\[ \int \left( e^{-x} \frac{dt}{dx} - e^{-x} t \right) dx = \int 1 \, dx \]
This simplifies to:
\[ e^{-x} t = x + C \]
where \( C \) is the constant of integration.

### Step 7: Solve for \( t \)
Substituting back for \( t \):
\[ e^{-x} (-\cos y) = x + C \]
Thus,
\[ -\cos y = (x + C)e^{x} \]
or
\[ \cos y = -(x + C)e^{x} \]

### Step 8: Use the Initial Condition
Using the initial condition \( y(0) = \frac{\pi}{2} \):
\[ \cos\left(\frac{\pi}{2}\right) = 0 \implies 0 = -(0 + C)e^{0} \implies C = 0 \]
So, we have:
\[ \cos y = -x e^{x} \]

### Step 9: Find \( y(1) \)
Now we need to find \( y(1) \):
\[ \cos y = -1 \cdot e^{1} = -e \]
Thus,
\[ y(1) = \cos^{-1}(-e) \]

### Final Answer
The value of \( y(1) \) is:
\[ y(1) = \cos^{-1}\left(-\frac{1}{e}\right) \]