What is the solution of the differential equation
`sin ((dy)/(dx))-a =0` ?

Options

A. `y =x sin ^(-1) a+c`
B. `x = y sin^(-1) a+c`
C. `y =x+x sin^(-1) a+c`
D. ` y = sin ^(-1) a +c`

Explanation

To solve the differential equation \( \sin\left(\frac{dy}{dx}\right) - a = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by isolating the sine function:
\[ \sin\left(\frac{dy}{dx}\right) = a \]

### Step 2: Taking the Inverse Sine
Next, apply the inverse sine function to both sides:
\[ \frac{dy}{dx} = \sin^{-1}(a) \]

### Step 3: Integrating Both Sides
Now, we integrate both sides with respect to \( x \):
\[ dy = \sin^{-1}(a) \, dx \]

### Step 4: Performing the Integration
Integrating the left side gives us \( y \), and integrating the right side gives us:
\[ y = \sin^{-1}(a) \cdot x + C \]
where \( C \) is the constant of integration.

### Final Solution
Thus, the solution to the differential equation is:
\[ y = x \sin^{-1}(a) + C \]

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