Solution of the diff. equation `sin((dy)/(dx))=a` when y(0) = 1

Options

A. `sin^(-1)[(y-1)/(x)]=a`
B. `sin[(y-1)/(x)]=a`
C. `sin*((1-y)/(1+x))=a`
D. `sin((y)/(x+1))=a`

Answer

To solve the differential equation \( \sin\left(\frac{dy}{dx}\right) = a \) with the initial condition \( y(0) = 1 \), we can 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 \( x \):
\[ \int dy = \int \sin^{-1}(a) \, dx \]
The left side integrates to \( y \), and the right side integrates to \( \sin^{-1}(a)x + C \), where \( C \) is the constant of integration:
\[ y = \sin^{-1}(a)x + C \]

### Step 3: Apply the initial condition
Now we apply the initial condition \( y(0) = 1 \):
\[ 1 = \sin^{-1}(a) \cdot 0 + C \]
This simplifies to:
\[ C = 1 \]

### Step 4: Write the final solution
Substituting \( C \) back into the equation gives us:
\[ y = \sin^{-1}(a)x + 1 \]

### Conclusion
Thus, the solution to the differential equation \( \sin\left(\frac{dy}{dx}\right) = a \) with the initial condition \( y(0) = 1 \) is:
\[ y = \sin^{-1}(a)x + 1 \]