`sin((dy)/(dx))=a`, when `x=0, y=1`

Options

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

Answer

To solve the differential equation \( \sin\left(\frac{dy}{dx}\right) = a \) with the initial condition \( x = 0, y = 1 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \sin\left(\frac{dy}{dx}\right) = a \]
To isolate \( \frac{dy}{dx} \), we take the inverse sine (arcsin) of both sides:
\[ \frac{dy}{dx} = \sin^{-1}(a) \]

### Step 2: Integrate both sides
Next, we integrate both sides with respect to \( x \):
\[ \int dy = \int \sin^{-1}(a) \, dx \]
Since \( \sin^{-1}(a) \) is a constant with respect to \( x \), we can integrate:
\[ y = \sin^{-1}(a) \cdot x + C \]
where \( C \) is the constant of integration.

### Step 3: Apply the initial condition
We use the initial condition \( x = 0, y = 1 \) to find \( C \):
\[ 1 = \sin^{-1}(a) \cdot 0 + C \]
This simplifies to:
\[ C = 1 \]

### Step 4: Write the particular solution
Now we substitute \( C \) back into the equation:
\[ y = \sin^{-1}(a) \cdot x + 1 \]

### Step 5: Solve for \( a \)
To express \( a \) in terms of \( y \) and \( x \), we rearrange the equation:
\[ y - 1 = \sin^{-1}(a) \cdot x \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ \sin^{-1}(a) = \frac{y - 1}{x} \]
Taking the sine of both sides gives us:
\[ a = \sin\left(\frac{y - 1}{x}\right) \]

### Final solution
Thus, the solution to the differential equation is:
\[ a = \sin\left(\frac{y - 1}{x}\right) \]

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