What is the solution of the differential equation sin `((dy)/(dx)) - a = 0` ?
Where C is an arbitrary constant

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 `

Answer

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

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

### Step 2: Apply the inverse sine function
To solve for \( \frac{dy}{dx} \), we take the inverse sine (arcsin) of both sides:
\[ \frac{dy}{dx} = \sin^{-1}(a) \]

### Step 3: Separate variables
Now, we can separate the variables by multiplying both sides by \( dx \):
\[ dy = \sin^{-1}(a) \, dx \]

### Step 4: Integrate both sides
Next, we integrate both sides:
\[ \int dy = \int \sin^{-1}(a) \, dx \]
Since \( \sin^{-1}(a) \) is a constant with respect to \( x \), the left side integrates to \( y \) and the right side integrates to:
\[ y = \sin^{-1}(a) \cdot x + C \]
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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